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THE 


COMPLETE 


AKITHMETIO 


ORAL    AND    WRITTEN. 


3t  DANIEL  W.   FISH,   A.M., 

EDITOR   OP    ROBINSON'S    SERIES    OF    PROGRESSIVE    ARITHMETICS. 


IVISON,  BLAKEMAN,  TAYLOR  &  CO., 
NEW  YORK   AND  CHICAGO 
1880. 


ROBINSON'S 

Shorter  Course 


FIRST  BOOK  IN-  ARITHME  TIC.  Primary. 
COMPLETE  ARITHMETIC.  In  One  w  7ume* 
COMPLETE  ALGEBRA. 

ARITHMETICAL  PROBLEMS.    Oral  and  Written. 
ALGEBRAIC  PROBLEMS.  ' 

KEYS  to  Complete  Arithmetic  and  Problems,  and 

to  Complete  Algebra  and  Problems, 

in  separate  volumes,  for  Teachers. 


Arithmetic,  orai  and  written,  usually  taught  in 
three  books,  is  now  offered,  complete  and  thorough, 
in  one  booh,  "  the  complete  arithmetic:- 

*  This  Complete  Arithmetic  is  also  published  in  two  volumes.  PART  I. 
and  PART  II.  are  each  bound  separately,  and  in  cloth. 


Copyright,  1874,  by  DANIEL  W.  FISH. 


Eltctrotyped  by  Smith  &  McDougal,  8a  Beekman  St.,  N.  Y. 


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RE FACE 


r  J^HE  design  of  the  author,  in  the  preparation  of  this  work,  has 
been  to  furnish  a  text-book  on  the  subject  of  arithmetic,  com- 
plete not  only  as  a  treatise,  but  as  a  comprehensive  manual  for  the 
class-room,  and,  therefore,  embodying  every  necessary  form  of 
illustration  and  exercise,  both  oral  and  written.  Usually,  this  rab~ 
ject  has  been  treated  in  such  a  way  as  to  form  the  contents  of  three 
or  more  graded  text-books,  the  oral  exercises  being  placed  in  a 
separate  volume.  In  the  present  treatise,  however,  the  whole 
subject  is  presented  in  all  its  different  grades  ;  and  the  oral,  or  men- 
tal, arithmetic,  so  called,  has  been  inserted,  where  it  logically  and 
properly  belongs,  either  as  introductory  to  the  enunciation  of  prin- 
ciples or  to  the  statement  of  practical  rules — the  treatment  of  every 
topic  from  the  beginning  to  the  end  of  the  book  being  thoroughly 
inductive. 

In  this  way,  and  by  carefully  constructed  analyses,  applied  to  all 
the  various  processes  of  mental  arithmetic,  the  pupil's  mind  cannot 
fail  to  become  thoroughly  imbued  with  clear  and  accurate  ideas  in 
respect  to  each,  particular  topic  before  he  is  required  to  learn,  or 
apply  to  written  examples,  any  set  rule  whatever.  The  intellect  of 
the  pupil  is  thus  addressed  at  every  step ;  and  every  part  of  the 
instruction  is  made  the  means  of  effecting  that  mental  development 
which  constitutes  the  highest  aim,  as  well  as  the  most  important 
result,  of  every  branch  of  education. 

This  mode  of  treatment  has  not  only  the  advantage  of  logically 
training  the  pupil's  mind,  and  cultivating  his  powers  of  calculation, 
but  must  also  prove  a  source  of  economy,  both  of  time  and  money, 
inasmuch  as  it  is  the  means  of  substituting  a  single  volume  for  an 
entire  series  of  text-books. 


M289984 


IV  PREFACE. 

As  the  time  of  many  pupils  will  not  permit  them  to  pursue  this 
study  through  all  of  its  departments,  the  work  is  issued  in  two  parts, 
as  well  as  in  a  single  volume.  This  will,  it  is  thought,  be  also  con- 
venient for  graded  schools,  in  supplying  a  separate  book,  for  classes 
of  the  higher  and  lower  grades  respectively,  without  requiring  any 
unnecessary  repetition  or  review. 

The  author  feels  assured  that,  on  examination,  this  work  will 
commend  itself  to  teachers  and  others,  by  the  careful  and  progres- 
sive grading  of  its  topics,  the  clearness  and  conciseness  of  its  defi- 
nitions and  rules,  its  improved  methods  of  analysis  and  operation, 
and  the  great  number  and  variety  of  its  progressively  arranged 
examples,  both  oral  and  'written,  embodying  and  elucidating  all  the 
ordinary  business  transactions.  The  use  of  equations  as  a  form  of 
expression  in  these  examples  will  be  found  to  possess  many  advan- 
tages, not  only  as  an  arithmetical  drill,  but  also  in  familiarizing  the 
pupil  with  the  use  of  algebraic  symbols. 

All  obsolete  terms  and  discarded  usages  have  been  studiously 
Ignored,  and  many  novel  features  introduced,  favorable  to  clearness 
as  well  as  brevity.  The  work  has  been  carefully  adapted,  in  other 
respects,  to  the  present  time,  recognizing  and  explaining  all  the 
recent  changes  in  Custom-house  Business,  Exchange,  etc.,  and  pre- 
senting, in  connection  with  the  examples  for  practice  under  eacli 
topic,  information  not  only  fresh  but  important. 

Attention  is  especially  called  to  the  manner  in  which  United 
States  Money  is  introduced  in  connection  with  the  elementary  rules  ; 
to  the  comprehensive  treatment  of  all  the  various  departments  of 
Percentage,  so  essential  at  the  present  time  in  commercial  transac 
tions ;  to  the  articles  on  Measurements  and  Mensuration,  and  the 
vast  amount  of  valuable  information  given  in  connection  with  this 
part  of  the  subject.  In  these  respects  this  part  of  the  work  will  be 
found  to  be  particularly  adapted  to  the  wants  of  High  Schools  and 
Academies,  as  well  as  of  Mercantile  and  Commercial  Colleges. 

The  Reviews  interspersed  throughout  the  book  will  be  found  to 
be  j  ust  what  is  needed  by  the  student  to  make  his  progress  sure 
at  each  step,  and  to  give  him  comprehensive  ideas  of  the  subject 
as  he  advances.    Carefully  constructed  Synopses  have  also  been 


PREFACE.  V 

inserted,  with  the  view  to  afford  to  both  teacher  and  pupil  a  ready 
means  of  drill  and  examination,  as  well  as  to  present,  in  a  clear, 
concise,  and  logical  manner,  the  relations  of  all  the  different  depart- 
ments of  the  subject,  with  their  respective  sub-topics,  defi?iitions, 
principles,  and  rules. 

Great  pains  have  also  been  taken  to  make  this  work  superior  to 
all  others  in  its  typographical  arrangement  and  finish,  and  in  the 
general  tastef  ulness  of  its  mechanical  execution. 

The  author  takes  pleasure  in  acknowledging  his  indebtedness  for 
many  valuable  suggestions  received  from  teachers  of  experience 
and  others  interested  in  the  work  of  education ;  especially  to  Joseph 
Ficklin,  Ph.  D.,  Professor  of  Mathematics  in  the  University  of 
Missouri,  by  whom  chiefly  the  sections  upon  Involution,  Evolution, 
Progressions,  and  Annuities  have  been  prepared  ;  as  well  as  to 
Henry  Kiddle,  A.  M.,  Superintendent  of  Schools  in  the  city  of  New 
York,  for  valuable  assistance,  especially  in  the  higher  departments 
of  Percentage,  and  for  important  suggestions  in  relation  to  other 
parts  of  the  work. 

How  nearly  the  author  has  accomplished  his  purpose,  to  give  to 
the  public,  in  one  volume,  a  clear,  scientific,  and  complete  treatise  on 
this  subject,  combining  and  systematizing  many  real  improvements 
of  practical  value  and  importance  to  the  business  man  and  the 
student,  the  intelligent  and  experienced  educator  must  decide. 

D.  W.  F. 

Bbooklyn,  January^  1876. 


T  N  order  to  teach  any  subject  with  the  best  success,  the  instruc- 
-*-  tor  should  not  only  fully  understand  it,  in  all  its  principles  and 
details,  but  should  also  clearly  perceive  what  particular  faculties 
of  the  mind  are  concerned  in  its  acquisition  and  use. 

Arithmetic  is  pre-eminently  a  subject  of  practical  value ;  that  is, 
it  is  one  to  be  constantly  applied  to  the  practical  affairs  of  life. 
But  this  is  true  only  in  a  limited  sense.  Very  few  ever  need  to 
apply  to  any  of  the  purposes  of  business  more  than  a  small  part  of 
the  principles  and  rules  of  calculation  taught  in  the  text-books. 
Every  branch  of  business  has  its  own  requirements  in  this  respect, 
and  these  are  all  confined  within  very  narrow  limits. 

The  teaching  of  arithmetic  must,  therefore,  to  a  great  extent,  be 
considered  as  disciplinary, — as  training  and  developing  certain 
faculties  of  the  mind,  and  thus  enabling  it  to  perform  its  functions 
with  accuracy  and  dispatch.  The  following  suggestions,  having 
reference  to  this  twofold  object  of  arithmetical  instruction  are  pre- 
sented to  the  teacher,  as  a  partial  guide,  not  only  in  the  use  of  this 
text-book,  but  in  the  treatment  of  the  subject  as  a  branch  of 
education. 

Seek  to  cultivate  in  the  pupil  the  habit  of  self-reliance.  Avoid 
doing  for  him  anything  which,  either  with  or  without  assistance, 
he  should  be  able  to  do  for  himself.  Encourage  and  stimulate  his 
exertions,  but  do  not  supersede  them. 

Never  permit  him  to  accept  any  statement  as  true  which  he  does 
not  understand.  Let  him  learn  not  by  authority  but  by  demonstra- 
tion addressed  to  his  own  intelligence.  Encourage  him  to  ask 
questions  and  to  interpose  objections.  Thus  he  will  acquire  that 
most  important  of  all  mental  habits,  that  of  thinking  for  himself. 


SUGGESTIONS     TO     TEACHERS.  Yll 

Carefully  discriminate,  in  the  instruction  and  exercises,  as  tc 
which  faculty  is  addressed, — whether  that  of  analysis  or  reasoning, 
or  that  of  calculation.  Each  of  these  requires  peculiar  culture,  and 
each  has  its  appropriate  period  of  development.  In  the  first  stage 
of  arithmetical  instruction,  calculation  should  be  chiefly  addressed, 
and  analysis  or  reasoning  employed  only  after  some  progress  has 
been  made,  and  then  very  slowly  and  progressively.  A  young 
child  will  perform  many  operations  in  calculation  which  are  far 
beyond  its  powers  of  analysis  to  explain  thoroughly. 

In  the  exercise  of  the  calculating  faculty,  the  examples  should  be 
rapidly  performed,  without  pause  for  explanation  or  analysis  ;  and 
they  should  have  very  great  variety,  and  be  carefully  arranged 
so  as  to  advance  from  the  simple  and  rudimental  to  the  complicated 
and  difficult. 

In  the  exercise  of  the  analytic  faculty,  great  care  should  be  taken 
that  the  processes  do  not  degenerate  into  the  mere  repetition  of 
formula.  These  forms  of  expression  should  be  as  simple  and  con- 
cise as  possible,  and  should  be,  as  far  as  practicable,  expressed  in 
the  pupil's  own  language.  Certain  necessary  points  being  attended 
to,  the  precise  form  of  expression  is  of  no  more  consequence  than 
any  particular  letters  or  diagrams  in  the  demonstration  of  geomet- 
rical theorems.  Of  course,  the  teacher  should  carefully  criticise 
the  logic  or  reasoning,  not  so  as  to  discourage,  but  still  insisting 
upon  perfect  accuracy  from  the  first. 

The  oral  or  mental  arithmetic  should  go  hand  in  hand  with  the 
written.  The  pupil  should  be  made  to  perceive  that,  except  for  the 
difficulty  in  retaining  long  processes  in  the  mind,  all  arithmetic 
ought  to  be  oral,  and  that  the  slate  is  only  to  be  called  into  requi- 
sition to  aid  the  mind  in  retaining  intermediate  processes  and 
results.  The  arrangement  of  this  text- book  is  particularly  favora- 
ble for  this  purpose. 

Definitions  and  principles  should  be  carefully  committed  to 
memory.  No  slovenliness  in  this  respect  should  be  permitted.  A 
definition  is  a  basis  for  thought  and  reasoning,  and  every  word 
which  it  contains  is  necessary  to  its  integrity.  A  child  should  not 
be  expected  to  frame  a  good  definition.     Of  course,  the  pupil  should 


Viii  SUGGESTIONS     TO     TEACHEES. 

be  required  to  examine  and  criticise  the  definitions  given,  since  this 
will  conduce  to  a  better  understanding  of  their  full  meaning. 

In  conducting  recitations,  the  teacher  should  use  every  means 
that  will  tend  to  awaken  thought.  Hence,  there  should  be  great 
variety  in  the  examples,  both  as  to  their  construction  and  phrase- 
ology, so  as  to  prevent  all  mechanical  ciphering  according  to  fixed 
methods  and  rules. 

The  Bules  and  Formula  given  in  this  book  are  to  be  regarded  as 
summaries  to  enable  the  pupil  to  retain  processes  previously  ana- 
lyzed and  demonstrated.  They  need  not  be  committed  to  memory, 
since  the  pupil  will  have  acquired  a  sufficient  knowledge  of  the 
principles  involved  to  be  able,  at  any  time,  to  construct  rules,  if  he 
has  properly  learned  what  precedes  them. 

In  the  higher  department  of  arithmetic,  the  chief  difficulty  con- 
sists in  giving  the  pupil  a  clear  idea  of  the  nature  of  the  business 
transactions  involved.  The  teacher  should,  therefore,  strive  by 
careful  elucidation,  to  impart  clear  ideas  of  these  transactions  before 
requiring  any  arithmetical  examples  involving  them  to  be  per- 
formed. When  the  exact  nature  of  the  transaction  is  understood, 
the  pupil's  knowledge  of  abstract  arithmetic  will  often  be  sufficient 
to  enable  him  to  solve  the  problem  without  any  special  rule. 

The  teacher  should  be  careful  not  to  advance  too  rapidly.  The 
mind  needs  time  to  grasp  and  hold  firmly  every  new  case,  and  then 
additional  time  to  bring  its  new  acquisition  into  relation  with  those 
preceding  it.  Hence  the  need  of  frequent  reviews,  in  order  to  give 
the  pupil  a  comprehensive  as  well  as  an  accurate  and  permanent 
knowledge  of  this  subject. 

The  Synopses  for  Review  interspersed  throughout  this  work  are 
designed  for  this  purpose.  The  whole  or  a  part  of  a  Synopsis, 
embracing  one  or  more  topics,  may  be  placed  upon  the  blackboard, 
and  the  pupil  required  to  give  briefly  but  accurately  the  subdivisions, 
definitions,  principles,  etc.,  involved  in  each.  By  this  means,  if 
further  tested  by  questions,  a  thorough  and  well  classified  know- 
ledge of  the  whole  subject  will  be  permanently  impressed  upon 
his  mind. 


gftniTOpKmfe 


FAGE 

Preliminary  Definitions 1 

Notation  and  Numeration 3 

Arabic  Notation 4 

Synopsis 12 

Addition 13 

Synopsis 22 

Subtraction 23 

Synopsis 34 

Multiplication.  35 

Synopsis 50 

Division 51 

Synopsis 76 

Properties  of  Numbers 77 

Divisibdlity  op  Numbers 79 

Factoring  80 

Common  Divisors 83 

Multiples 88 

Cancellation 92 

Synopsis 96 

Fractions 97 

Definitions 98 

General  Principles 100 

Reduction 101 

Addition 109 

Subtraction 112 

Multiplication 115 

DmsioN 124 

Relation  of  Numbers 132 

Synopsis 141 

Decimals 142 


PAGE 

Notation  and  Numeration 144 

Decimal  Currency . . .  150 

Reduction 151 

Addition 156 

Subtraction 158 

Multiplication 159 

DmsioN 162 

Circulating  Decimals 165 

Short  Methods 169 

Ledger  Accounts 175 

Accounts  and  Bills 176 

Synopsis 183 

Denominate  Numbers 184 

Measures  of  Extension 186 

Measures  of  Capacity 191 

Measures  of  Weight 194 

Synopsis 198 

Measures  of  Time 199 

Measures  of  Angles ,  200 

Miscellaneous  Measures 203 

Measures  of  Value 204 

Synopsis' 208 

Reduction  of  Denom.  Integers..  £09 
Reduction  of  Denom.  Fractions.  21(5 

Addition 225 

Subtraction 227 

Multiplication 230 

Division 231 

Longitude  and  Time.... 233 

Duodecimals 239 


CONTENTS 


PAGE 

Synopsis 240 

Measurements— Surfaces 241 

Land 245 

Rectangular  Solids 249 

Boards  and  Timber 254 

Capacity  of  Bins,  Cisterns,  Etc.  257 

Synopsis 264 

Percentage 265 

Profit  and  Loss 276 

Commission 234 

Synopsis 292 

Interest. 293 

Problems  in  Interest 304 

Compound  Interest 309 

Annual  Interest 312 

Partial  Payments 314 

Discount 318 

Savings  Banks 326 

Synopsis 329 

Stocks 330 

Insurance 340 

Life  Insurance 344 

Taxes 348 

Synopsis 352 

Exchange 353 

Arbitration  of  Exchange 362 

Custom-house  Business  366 

Equation  of  Payments     .   369 

Averaging  Accounts 374 


Synopsis 382 

Ratio .  383 

Proportion 387 

Partnership 401 

Alligation 407 

Synopsis 413 

Involution 419 

Evolution 425 

Progressions ...  439 

Annuities 449 

Synopsis 453 

Mensuration 454 

Triangles 455 

Quadrilaterals 459 

Circles 461 

Similar  Plane  Figures 464 

Solids 467 

Prisms 467 

Pyramids  and  Cones 469 

Spheres 472 

Similar  Solids 473 

Gauging 474 

Synopsis 476 

Metric  System 477 

Vermont  Partial  Payments 491 

Vermont  Taxes 495 

Tables 497 

Answers 499 


N.  B. — Editions  of  this  book  arc  bound  with,  and  icithout,  the 
answers.  The  edition  with  answers  will  be  supplied  unless  other- 
wise ordered. 


A^K> 


5^A  < 


ARTICLE  1.  Arithmetic  is  the  Science  of  Num- 
-*-^-  bers,  and  the  Art  of  Computation. 

As  a  science,  Arithmetic  treats  of  the  nature  and  properties  of 
numbers.  As  an  art,  it  teaches  how  to  apply  a  knowledge  of  num- 
bers to  practical  and  business  purposes. 

2.  A  Unit  is  one,  or  a  single  thing  ;  as  one,  one  boy, 
one  year,  one  dozen. 

3.  A  Number  is  a  unit,  or  a  collection  of  units  ;  as 
one,  three, five  boys;  it  answers  the  question,  How  many? 

4.  An  Integral  Number  or  Integer  is  a  num- 
ber which  expresses  whole  things  ;  as  seven,  four  days. 

5.  The  Unit  of  a  Number  is  one  of  the  collection 
of  units  which  constitute  the  number.  Thus,  the  unit 
of  twelve  is  one,  of  twenty  dollars  is  one  dollar. 

6.  A  Concrete  Number  is  a  number  that  is  applied 
to  a  particular  kind  of  object,  or  quantity;  as  three 
houses,  four  dollars,  five  minutes. 

7.  An  Abstract  Number  is  a  number  that  is  not 
applied  to  any  object ;  as  four,  seven,  eight 


2  DEFINITIONS. 

8.  Like  Numbers  are  such  as  have  the  same  kind 
of  unit,  or  express  the  same  kind  of  quantity.  They  may 
be  either  concrete  or  abstract ;  as  eight  and  nine,  six  days 
and  ten  days,  two  rods  five /ee£,  and  five  rods  three  feet. 

9.  Tinlike  Numbers  are  such  as  have  different 
kinds  of  units,  or  express  different  kinds  of  quantity  ;  as 
ten  months  and  eight  miles  ;  seven  dollars  and  five  barrels. 

10.  A  Scale  in  Arithmetic,  is  a  succession  of  units, 
increasing  and  decreasing  according  to  a  certain  law,  or 
rule.     Scales  are  uniform  or  varying. 

11.  A  Uniform  Seale  is  one  in  which  the  law  of 
increase  and  decrease  is  the  same  throughout  the  entire 
succession  of  units. 

12.  A  Varying  Seale  is  one  in  which  the  law  of 
increase  and  decrease  is  not  the  same  throughout  the 
entire  succession  of  units. 

13.  A  Decimal  Scale  is  one  in  which  the  law  of 
increase  and  decrease  is  uniformly  ten. 

EXER  CISES. 

14.  1.  How  many  units  in  two  ?  In  five  cents  ?  In 
six  dollars  ?    In  seven  acres  ? 

2.  What  is  the  unit  of  six  cents  ?     Of  nine  books  ? 

3.  Are  two  trees  and  five  trees  like  or  unlike  numbers  ? 

4.  Are  they  concrete  or  abstract  ?    Why  ? 

5.  What  kind  of  numbers  are  seven  and  nine  ?  Are 
fWe  acres  and  seven  cords  ?    Are  four  coats  and  six  coats  ? 

6.  Name  two  numbers  that  are  like  and  abstract. 

7.  Name  two  numbers  that  are  like  and  concrete. 

8.  Name  three  numbers  that  are  unlike  and  concrete. 


NOTATION     A^D     NUMERATION, 


One  Thousand. 


Hundred  and  Eleven. 


NOTATION  AND   NUMERATION. 

15.  In  representing  numbers,  objects  are  regarded  as 
arranged  in  groups  of  tens  ;  hence  we  have  single  things, 
or  units;  next,  groups  containing  ten  units,  or  ten  ;  next, 
groups  containing  ten  tens,  or  one  hundred ;  and  again, 
groups  containing  ten  hundreds,  or  one  thousand,  etc. 

16.  This  method  of  grouping  is  called  the  Decimal 
System,  from  the  Latin  word  decern,  which  signifies  ten. 

IK.  Notation  is  a  method  of  writing,  or  represent- 
ing numbers  by  characters. 

18.  Numeration  is  a  method  of  reading  numbers 
represented  by  characters. 


*  DOTATION     AND     NUMERATION/. 

19.  The  number  of  objects  may  be  represented  by 
words,  or  by  characters. 

20.  The  characters  may  be  either  figures  or  letters. 

21.  Figures  are  characters  used  to  express  number?. 

22.  The  Arabic  Notation  is  the  method  of  ex- 
pressing numbers  by  figures.  It  is  so  called  because  it 
was  invented  by  the  Arabs. 

23.  This  method  employs  ten  different  characters,  or 
figures,  to  represent  numbers,  viz.  : 

Figures.   0,    1,   2,    3,    4,   5,   6,    7,    8,    9. 

JSameS,     Naught,  One,  Two,  Three,  Four,  Five,  Six,  Seven,  Eight,  Nine. 

24.  The  first  character,  or  cipher,  is  called  Naught,  or 
Zero,  and  when  standing  alone,  has  no  value. 

The  other  nine  are  called  significant  figures,  because 
each  has  a  value  of  its  own.     They  are  also  called  digits. 

These  ten  characters,  when  combined  according  to  cer- 
tain principles,  can  be  made  to  express  any  number. 

25.  The  first  nine  numbers  are  each  represented  by  a 
single  figure,  and  are  called  units  of  the  first  order. 

26.  By  grouping  ten  ones,  or  units  of  the  first  order 
into  a  larger  collection,  there  is  formed  a  unit  of  the 
second  order,  called  ten,  which  is  represented  by  writing 
the  figure  1  with  a  cipher  after  it ;  thus,  10. 

27.  In  the  same  manner  are  represented 


Two  tens,  or  Twenty,  by  20 
Three  tens,  or  Thirty,  "  30 
Four  tens,  or  Forty,  "  40 
Five  tens,  or  Fifty,      "  50 


Six  tens,  or  Sixty,  by  GO 
Seven  tens,  or  Seventy, "  70 
Eight  tens,  or  Eighty,  "  80 
Nine  tens,  or  Ninety,  "  90 


NOTATION"     AND     NUMERATION 


28.  The  numbers  between  ten  and  twenty  are  repre- 
sented by  writing  1  in  the  second  place,  and  the  units  in 
the  first  place.     Thus, 


Eleven 
Twelve 
Thirteen 


11 
12 
13 


Fourteen 

Fifteen 

Sixteen 


14 
15 
16 


Seventeen 

Eighteen 

Nineteen 


17 
18 
19 


29.  In  like  manner,  the  numbers  between  20  and  30 
are  represented,  thus, 


Twenty-one  21 
Twenty- two  22 
Twenty-three  23 


Twenty-four  24 
Twenty-five  25 
Twenty-six      26 


Twenty-seven  27 
Twenty-eight  28 
Twenty-nine     29 


30.  The  greatest  number  that  can  be  expressed  by 
two  figures  is  99. 

31.  By  grouping  ten  units  of  the  second  order,  or  ten 
tens,  into  a  larger  collection,  there  is  formed  a  unit  of 
the  third  order,  called  a  hundred,  represented  by  writing 
the  figure  1  with  two  ciphers  after  it ;  thus,  100. 

32.  In  like  manner  are  represented 


Two  hundred 

by  200 

Six  hundred        by  600 

Three  hundred 

"  300 

Seven  hundred     "   700 

Four  hundred 

"   400 

Eight  hundred     "   800 

Five  hundred 

"  500 

Nine  hundred       "  900 

33.  The  numbers  from  one  hundred,  to  nine  hundred 
and  ninety-nine,  are  represented  by  writing  the  hundreds 
in  the  third  place,  the  tens  in  the  second  place,  and  the 
units  in  the  -first  place. 

34.  The  greatest  number  that  can  be  expressed  by 
three  figures  is  999. 


6  NOTATION     AND     NUMERATION. 

35.  Orders  of  Units  are  denoted  by  the  position 
of  the  figures  used  in  expressing  a  number. 

Thus,  532  represents  2  units  of  the  first  order,  3  units  of  the 
second  order,  or  3  tens,  and  5  units  of  the  third  order,  or  5  hundreds, 
and  is  read  five  hundred  and  thirty-two. 

36.  Principles. — 1.  Ten  units  of  any  order  in  a 
number  make  one  unit  of  the  next  higher  order. 

2.  When  any  order  of  units  in  a  number  is  vacant,  the 
place  is  filled  with  a  cipher, 

EXERCISES. 

37.  Express  the  following  numbers  by  figures  : 

7.  Six  hundred  ninety. 

8.  Eight  hundred  five. 

9.  Seven  hundred  ten. 


1.  One  hundred  twenty. 

2.  Four  hundred  eighty. 

3.  Seven  hundred  six. 

4.  Five  hundred  seven. 

5.  Seven  hundred. 

6.  Three  hundred  eight. 


10.  Six  hundred  eleven. 

11.  Nine  hundred  seven. 

12.  .Two  hundred  sixty. 


38.  Copy  and  read  the  following,  and  name  the  num- 
ber of  hundreds,  tens,  and  units  in  each  : 


(1.) 

(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

67 

321 

190 

840 

592 

219 

85 

406 

761 

269 

904 

807 

77 

289 

345 

793 

531 

395 

98 

672 

402 

503 

762 

608 

39.  By  grouping  ten  units  of  the  third  order,  or  ten 
hundreds,  into  a  larger  collection,  there  is  formed  a  unit 
of  the  fourth  order,  called  a  thousand,  represented  by 
writing  the  figure  1  with  three  ciphers  after  it ;  thus,  1000. 


NOTATION     AND     NUMEKATION. 


40.  In  like  manner  are  represented 


Two  thousand  by  2000 
Three  thousand  u  3000 
Four  thousand  "  4000 
Five  thousand      '•   5000 


Six  thousand  by  6000 

Seven  thousand  "   7000 

Eight  thousand  u  8000 

Nine  thousand  "  9000 


41.  The  numbers  from  one  thousand,  to  nine  thousand 
nine  hundred  and  ninety-nine,  are  represented  by  writing 
thousands  in  the  fourth  place,  hundreds  in  the  third  place, 
tens  in  the  second  place,  and  units  in  the  first  place. 

Thus,  5304  represents  4  units  of  the  first  order,  0  units  of  the 
second  order,  or  tens,  3  units  of  the  third  order,  or  hundreds,  and 
5  units  of  the  fourth  order,  or  thousands,  and  is  read  five  thousand 
three  hundred  and  four. 

42.  The  greatest  number  that  can  be  expressed  by  four 
figures  is  9999. 

43.  In  the  same  manner,  other  new  orders  are  formed 
to  represent  larger  numbers,  by  grouping  ten  units  of  the 
fourth  order  to  form  the  fifth  order,  or  tens  of  thousands  ; 
and  ten  units  of  the  fifth  order,  to  form  the  sixth  order, 
or  hundreds  of  thousands,  etc. 

Thus,  432076  represents  6  units  of  the  first  order,  7  units  of 
the  second  order,  0  units  of  the  third  order,  2  units  of  the  fourth 
order,  3  units  of  the  fifth  order,  and  4  units  of  the  sixth  order,  and 
is  read  four  hundred  thirty-two  thousand  and  seventy-six. 

From  the  preceding  illustrations  it  is  obvious,  that 

44.  Moving  a  figure  one  place  to  the  left,  increases  its 
representative  value  tenfold;  and, 

45.  Moving  a  figure  one  place  to  the  right,  diminishes 
its  representative  value  tenfold. 


8  NOTATION     AND     NUMERATION. 

EXERCISES. 

46.  "Write  in  figures  and  read  : 

1.  Two  units  of  the  third  order,  four  units  of  the 
second  order,  and  three  units  of  the  first  order. 

2.  Five  units  of  the  fourth  order,  six  units  of  the  third 
order,  and  two  units  of  the  second  order. 

3.  Seven  units  of  the  fourth  order,  eight  of  the  second 
order,  and  three  of  the  first. 

4.  One  unit  of  the  third  order  and  four  of  the  second. 

5.  Three  units  of  the  fifth  order,  two  of  the  third,  and 
one  of  the  first. 

6.  Eight  units  of  the  fourth  order,  and  five  of  the 
second. 

7.  Two  units  of  the  sixth  order,  nine  of  the  fifth,  four 
of  the  third,  one  of  the  second,  and  seven  of  the  first. 

47.  Express  the  following  numbers  by  figures  : 

1.  Thirty-seven  thousand. 

2.  Sixteen  thousand  one  hundred. 

3.  Twelve  thousand  five  hundred  fifty. 

4.  Forty-nine  thousand  five  hundred  twenty-seven. 

5.  Fifteen  thousand  two  hundred  six. 

6.  Seventeen  thousand  twenty-four. 

7.  Sixty  thousand  six  hundred  eight. 

8.  Seven  hundred  twenty  thousand. 

9.  Two  hundred  forty  thousand  five  hundred. 

48.  Copy,  and  read  the  following,  naming  the  num- 
ber of  units  of  each  order : 

(1.)  (2.)  (3.)  (4.)  (5.) 

1542         1020         32507  76387         528031 

3473         1256         53106        627324         600320 


NOTATION     AND     NUMERATION. 


49.  This  method  of  numeration  groups  the  successive 
orders  into  periods  of  three  figures  each.  The  periods 
are  commonly  separated  by  commas,  each  period  taking 
the  name  of  its  lowest  order,  as  shown  in  the  following 


Periods. 


Name. 


Orders 

of  Units 

IN  THE 

Periods. 


Numeration  Table. 
6th.       5th.       4th.        3d. 


3 


.2 
H 


PQ 


2d. 


-3 
d 

a 

m 
P 

O 

pd 


1st. 


£ 


§ 


00 

CD 

H 

p  fl 


no 
g 

«3 


CO 

'd 

d 


DQ 

•73 


'2 
P 
3  0  6 


CO 

H 

•"d 
d    M 
P 


-  g'd 
4  0  0 


INTHE  g2.-S  gg.t3  d  g.-S  d  g.-d  d  g 
Periods.  £S£  ^H^  k££  WHP  «£ 
Number.        30,291,040,027,30 

The  number  is  read  30  quadrillion,  291  trillion,  40  billion,  27  ra#- 
#<ra,  30G  t/iousand,  400. 

1.  In  reading  numbers,  the  name  of  the  units  period  is  omitted. 

2.  Every  period  except  the  highest  must  contain  three  figures. 


50. 

The  names  of  the  periods  above  Quadrillions  are 

Periods. 

Names. 

Periods. 

Names. 

7th 

Quintillions. 

15th 

Tredecillions. 

8th 

Sextillions. 

16th 

Quatuord  ecillions. 

9th 

Septillions. 

17th 

Quindecillions. 

10th 

Octillions. 

18th 

Sexdecillions. 

11th 

Nonillions. 

19th 

Septendecillions. 

12th 

Decillions. 

20th 

Octodecillions. 

13th 

Undecillions. 

21st 

Novendecillions. 

14th 

Duodecillions. 

22d 

Vigintillions. 

10 


NOTATION     AND     NUMERATION. 


■§ 

a 

KEHP 

WHt^ 

HEh'P 

tifr+P 

3  4  1 

12  5 

0  0  4 

4 

0  4  4 

0  3  4 

3  3 

3  0  0 

3  3  0 

2  0  0 

2  2  0 

0  2  2 

5 

0  0  5 

5  0  0 

0  5  5 

51.  The  pupil  may  be 
required  to  prepare  and 
arrange  on  the  slate  or  on 
paper,  exercises  similar  to 
the  following. 

The  first  example  is 
read,  341. 

The  second,  is  read, 
125  thousand  and  4. 

The  third,  is  read,  4 
million  44  thousand  and  34. 

The  fourth,  is  read,  33  million  300  thousand  330. 

The  diagram  may  be  prepared  at  first  for  only  two  or  three 
periods,  and  may  be  gradually  enlarged  to  five  or  six  periods. 

Each  pupil  may  be  allowed  to  dictate  ■  an  example,  to  be  written 
and  read  by  the  whole  class. 

52.  Eule  for  Notation. — Begin  at  the  left,  and 
write  the  hundreds,  tens,  and  units  of  each  period  in  their 
proper  order,  filling  all  vacant  places  and  periods  with 
ciphers. 

53.  Eule  for  Numeration. — I.  Begin  at  the  right, 
and  separate  the  number  into  periods  of  three  figures  each. 

II.  Begin  at  the  left,  and  read  each  period  as  if  it  were 
units,  giving  its  name. 

EXERCISES    IN    NOTATION    ANI>    NUMERATION. 

54.  Write  in  figures  and  read  : 

1.  Six  units  of  the  3d  order,  five  of  the  2d,  and  four 
of  the  1st. 

2.  Five  units  of  the  4th  order,  seven  of  the  2d,  and  six 
of  the  1st. 


NOTATION     AND     NUMERATION.  11 

3.  Eight  units  of  the  4th  order  and  four  of  the  2d. 

4.  Five  units  of  the  4th  order  and  eight  of  the  2d. 

5.  Three  units  of  the  5th  order,  six  of  the  4th,  four 
of  the  3d,  and  seven  of  the  1st. 

6.  Two  units  of  the  6th  order,  four  of  the  5th,  nine  of 
the  4th,  three  of  the  3d,  and  five  of  the  1st. 

7.  Three  units  of  the  9th  order,  eight  of  the  7th,  four 
of  the  6th,  six  of  the  5th,  and  nine  of  the  1st. 

55,  Write  and  read  the  following  numbers  in  figures : 

1 .  Twenty-five  units  in  the  2d  period,  and  four  hun- 
dred ninety-six  in  the  1st.  Ans.  25,496. 

2.  Four  hundred  thirty-six  units  in  the  4th  period,' 
twelve  in  the  3d,  one  hundred  in  the  2d,  and  three  hun- 
dred and  one  in  the  1st. 

3.  Eighty-one  units  in  the  5th  period,  two  hundred 
and  nineteen  in  the  4th,  and  fifty-six  in  the  2d. 

4.  Nine  hundred  and  forty  units  in  the  seventh  period, 
eighteen  in  the  fifth,  and  one  hundred  and  three  in  the  3d, 

56.  Express  the  following  numbers  by  figures  : 

1.  Twenty-six  thousand  twenty-six. 

2.  Fourteen  thousand  two  hundred  eighty. 

3.  One  hundred  seventy-six  thousand. 

4.  Four  hundred  fifty  thousand  thirty-nine. 

5.  Seven  million  thirty-six. 

6.  Five  hundred  sixty- three  thousand  four. 

7.  One  million  ninety-six  thousand. 

8.  Ten  million  ten  thousand  ten  hundred  ten. 

9.  Four  hundred  eighty-three  million  eight  hundred 
sixteen  thousand  one  hundred  forty- nine. 

10.  Ninety-nine  billion  thirty-seven  thousand  four. 


12  NOTATION     AND     N  U  MEK  ATI  0  N. 

Point  off  and  read  the  following  numbers : 

1.  24835.  5.     100103. 

2.  2474783.  6.    53000008. 

3.  31628045.  7.   406270035. 

4.  247843112.  8.  3730016000. 

55.  Roman  Notation  employs  seven  capital  letters 
to  express  numbers.     Thus, 

Letters.    I,       V,       X,        L,        O,        D,        M. 

Values.     1,       5,        10,       50,      100,      500,     1000. 

When  used  alone,  each  letter  has  its  fixed  value. 
Numbers  may  be  expressed  by  combining  these  letters 
according  to  the  following  principles: 

1.  Repeating  a  letter  repeats  its  value. 
Thus,  XX  represents  20 ;  CCC,  300 ;  DD,  1000. 

2.  When  a  letter  is  placed  after  one  of  greater  value,  its 
value  is  to  be  added  to  that  of  the  greater. 

Thus,  VI  represents  6 ;  XV,  15 ;  LXX,  70 ;  DC,  600. 

3.  When  a  letter  is  placed  before  one  of  greater  value, 
its  value  is  taken  from  that  of  the  greater. 

Thus,  IV  represents  4;  IX,  9  ;  XL,  40  ;  XC,  90. 

4.  When  a  letter  of  any  value  is  placed  between  two 
letters,  each  of  greater  value,  its  value  is  taken  from  the 
sum  of  the  other  two. 

Thus,  XIV  represents  14;  LIX,  59;  CXL,  140. 

5.  A  bar  or  dash  placed  over  a  letter  increases  its  value 
one  thousand  times. 

Thus,  X  represents  10000 ;  XC,  90000 ;  DL,  550000. 


NOTATION     AND 

NUM 

EEATIOS 

Via 

Table  of  Roman  Notation. 

I 

— 

l 

XVI 

= 

16 

C 

S3 

100 

II 

— 

2 

XVII 

= 

17 

CXIX 

r= 

119 

III 

= 

3 

XVIII 

= 

18 

cc 

= 

200 

IV 

— 

4 

XIX 

= 

19 

ccx 

= 

210 

V 

rr 

5 

XX 

- 

20 

D 

= 

500 

VI 

== 

6 

XXI 

■=. 

21 

DCV 

= 

605 

VII 

= 

7 

XXV 

= 

25 

M 

= 

1000 

VIII 

3= 

8 

XXX 

= 

30 

MDL 

= 

1550 

IX 

= 

9 

XXXIV 

= 

34 

MDCLXVI 

S3 

1666 

X 

= 

10 

XL 

= 

40 

MXCIX 

= 

1099 

XI 

— 

11 

L 

= 

50 

XXV 

= 

25000 

XII 

SB 

12 

LX 

= 

60 

CXX 

= 

120000 

XIII 

=r 

13 

LXX 

rr 

70 

CLXIV 

= 

164000 

XIV 

±s 

14 

LXXX 

= 

80 

DLCXL 

= 

550140 

XV 

=5 

15 

XC 

= 

90 

MDXC 

== 

1000590 

MDCCCLXXX  =  1880,  one  thousand  eight  hundred  and  eighty. 


56,  Express  by 

1.  Twenty-seven. 

2.  Forty-nine. 

3.  Seventy-three. 

4.  Sixty-eight. 

5.  Eighty-four. 

6.  Ninety-seven. 

57.  Express  by 

1.  LXVII.  6, 

2.  XCLXIV.      7. 

3.  CXXXV. 

4.  CCXLIX. 

5.  MXIX. 


EXERCISES, 

Roman  notation: 

7.  One  hundred  ten. 

8.  Five  hundred  fifty. 

9.  Seven  hundred  forty. 

10.  Nine  hundred  ninety. 

11.  Sixteen  hundred. 

12.  Fifty  thousand  five. 

Arabic  notation : 


13.  318. 

14.  796. 

15.  1069. 

16.  25000. 

17.  59300. 

18.  87040. 


DCLIII. 
OXCIX. 

8.  VDLIX. 

9.  DLX. 
10.  XXXID. 


11.  LIXCCCXLIV. 

12.  XVDCCXLIX. 

13.  MMMMXC. 

14.  VMDCCXLIX. 

15.  MDXXVCDLXXXIX. 


12b 


NOTATION     AND     NUMERATION 


58. 


SYNOPSIS    FOR    REVIEW. 


1.  Arithmetic.  2.  A  Unit.  3.  A  Number.  4.  An  In- 
tegral Number.  5.  Unit  of  a  Number.  6.  A  Concrete 
Number.  7.  An  Abstract  Number.  8.  Like  Numbers. 
9.  Unlike  Numbers.  10.  A  Scale.  11.  A  Uniform 
Scale.     12.  A  Varying  Scale.    13-  A  Decimal  Scale. 


1.  Definitions. 


2.  Mode  of  Rep- 
resenting 


r  1.  Decimal  System. 

I  2.  Notation. 

<j  3.  Numeration. 

I  4.  Figures. 

^  5.  Arabic  Notation. 

'  1.  Units  of  the  first    order. 

2.  "        "       second  " 

3.  "        "       third     " 

4.  *        "       fourth  " 

5.  Numbers  between  ten  and  twenty 

between  twenty  and  thirty. 

6.  Other  orders  and  numbers. 


3.  Order  of  Units— how  denoted. 

4.  Principles,  1  and  2. 


5.  Value    of 
Figures. 


6.  Periods. 


it 


How  increased. 
How  diminished. 


7.  Rules. 


1.  Of  how  many  figures. 

2.  How  separated. 

3.  Names  of  Periods  to    Quadrillions. 

4.  -       "        "       beyond    " 

1.  For  Notation. 

2.  For  Numeration. 


r  1.  How  expressed. 
8.  Roman  Nota-  j  2.  How  many  letters. 

tion.  I  3.  Principles  1 ,  2,  3,  4,  and  5. 

I  4.  Table. 


Kg 


^->©* 


OJtAE     EXERCISES. 

59.  1.  A  man  gave  5  dollars  for  a  hat,  and  9  dollars 
for  a  vest.     How  many  dollars  did  he  pay  for  both  ? 

2.  How  many  miles  are  4  miles  and  12  miles? 

3.  How  many  are  6  men  and  14  men  ?     15  and  7  ? 

4.  How  many  are  14  and  5  ?    29  and  5  ?    24  and  5  ? 

5.  How  many  are  15  and  6  ?    21  and  6  ?    27  and  6  ? 

6.  How  many  are  7  and  12  ?     7  and  19  ?     7  and  26  ? 

7.  What  kind  of  numbers  are  9  pounds  and  14  pounds  ? 

8.  Can  9  balls  and  12  books  be  added  ?    Why  not  ? 

9.  Can  7  rods  and  10  rods  be  added  ?    Why? 

10.  How  many  are  4,  5,  and  7?  5,  7,  and  4?  4,  7,  and  5? 

11.  In  a  shop  are  15  men,  8  boys,  and  6  girls  at  work. 
How  many  persons  are  at  work  in  the  shop  ? 

12.  I  gave  7  cents  to  one  boy,  9  to  another,  and  6  to 
another.     How  many  cents  did  I  give  to  all  ? 

13.  Add  by  2's  from  1  to  31. 

Operation.— 1,  3,  5,  7,  9,  11, 13,  15,  17, 19,  21,  23,  25,  27,  29,  31 


Add 

14.  By  2's  from  4  to  50. 

15.  By  3's  from  1  to  43. 

16.  By  3's  from  6  to  51. 

17.  By  4's  from  1  to  53. 

18.  By  4's  from  5  to  45. 


19.  By  5's  from    1  to  61. 

20.  By  5's  from    7  to  82. 

21.  By  6's  from    0  to  72. 

22.  By  6's  from    2  to  80. 

23.  By  6's  from  10  to  88. 


14 


ADDITION. 


24.  Add  by  2's  and  3's  alternately  from  1  to  31. 
Operation.— 1,  3,  6,  8,  11,  13,  16,  18,  21,  23,  26,  28,  31. 

25.  Add  by  2's  and  4's  alternately  from  1  to  49. 

26.  Add  by  3's  and  4's  alternately  from  0  to  56. 

27.  Add  by  2's  and  5's  alternately  from  4  to  60. 

28.  Add  by  3's  and  5's  alternately  from  7  to  63. 

DEFINITIONS    AND    PRINCIPLES. 

60.  Addition  is  the  process  of  finding  a  number 
equivalent  to  two  or  more  numbers. 

61.  The  Sum  or  Amount  is  the  number  obtained 
by  addition. 

62.  The  Sign  of  Addition  is  -f .  It  is  read  j^s, 
and  signifies  more;  thus,  5  -f-  6  is  read,  5  plus  6,  and 
means  that  5  and  6  are  to  be  added. 

63.  The  Sign  of  Equality  is  =.  It  is  read 
equals,  or  equal  to ;  thus,  5  +  6  =  11,  is  read  5  plus  6 
equals  11.     It  may  be  read  5  and  6  are  11. 

64.  An  Equation  is  an  expression  of  equality  be- 
tween two  numbers  or  sets  of  numbers. 

All  that  is  written  he- 
fore  the  sign  of  equality 
is  called  the  first  mem- 
ber of  the  equation,  all 
that  is  written  after  the 
sign  of  equality  is  called 
the  second  member. 

The  numbers  in  each 
member  are  called  the 
EQUATION.  terms  of  the  equation. 


ADDITION".  15 

Thus,  6  +  4  =  10,  is  an  equation,  and  is  read  6  plus  4  equals  10, 
and  means  that  the  sum  of  6  and  4  is  equal  to  10.  6  +  4  is  the 
first  member  of  the  equation,  and  10  is  the  second  member;  and 
6,  4,  and  10  are  the  terms  of  the  equation. 

65,  Name  the  members  and  the  terms  of  each  of  the 
following  equations. 


1.  9  +  12=21 

2.  14  +  10=24 


3.  26+   9=35 

4.  44  +  12  =  56 


5.  14  +  40=44  +  10 

6.  18  +  20=31+   7 


66.  Principle. — Only  like  numbers  and  units  of  the 
same  order  can  be  added. 

EXERCISES. 

67.  The  teacher  should  read  the  first  member  of  the 
equation,  and  the  pupil  be  required,  as  promptly  as  pos- 
sible, to  give  the  second  member. 

The  expression  "  =  ?"  is  read  equals  how  many,  or  what. 

6  +  4  +  5=?  3  +  12+   6=?  12  +  5  +  10=? 

7  +  0  +  2=?  6+4  +  10=?  15  +  2+4=? 

8  +  3  +  1=?  2  +  11+5=?  13  +  7+8=?. 

9  +  2  +  4=?  8+   9+   3=?  16  +  5+   6=  ? 
7  +  6  +  5=  ?  3  +  12+7=?  18  +  0  +  10=? 

10  +  8  +  3=?  14+   7+   6=?  20  +  6+   8=  ? 


5  +  8+   9=?  20+   7+  3=?  4  +  15  +  10=? 

9  +  8+  5=?  8  +  12  +  10=?  8+   8+   8=  ? 

8+  5+   9=?  11+   9+   1=?  9+   9+   9=? 

10+  7+   6=?  12+   7+   9=?  23  +  10+   6=  ? 

6+-10+   7=?  21  +  10+7=?  7  +  21  +  11=? 

H+  5+   8=?  24+   6+   5=?  14+   6  +  12=? 

15+  3  +  10=?  10  +  25+   9=?  26  +  10+   9  -  ? 


OPERATION. 

421 

44 

303 

230 

16  ADDITION. 


WRITTEN     EXERCISES, 

68.  When  the  sum  of  the  units  of  each  order  is 
less  than  lO. 

1.  What  is  the  sum  of  421,  44,  303,  and  230  ? 

Analysis. — Arrange  the  numbers  so  that  the 
units  of  the  same  order  stand  in  the  same  column. 
Begin  with  the  lowest  order  of  units,  and  add 
each  column  separately  ;  and  instead  of  saying,  3 
units  and  4  units  are  7  units,  and  1  unit  are  8  units, 
pronounce  the  successive  results  only ;  thus,  3,  7, 
9  9  8  Sum.     8>  the  sum  of  the  units,  which  write  in  the  units' 

place. 
Next,  3,  7,  9,  the  sum  of  the  tens,  which  write  in  the  tens'  place. 
Lastly,  2,  5,  9,  the  sum  of  the  hundreds,  which  write  in  the  hun- 
dreds' place.    Hence  the  sum  is  998. 

Proof. — Add  the  columns  in  the  reversed  direction.  If  the  two 
results  agree,  the  work  is  probably  correct. 

Copy,  add,  and  prove, 

(2.)                (3.)  (4.)  (5.) 

204                312  241  403 

462                243  520  2052 

_23                124  _27  4324 

6.  I  paid  3104  dollars  for  a  house,  450  dollars  for  repairs, 
and  234  dollars  for  painting.     What  was  the  whole  cost  ? 

69.  The  Sign  of  Dollars  is  $.  It  is  read  dollars. 
Thus,  $35  is  read  35  dollars  ;  $9  is  read  9  dollars. 

70.  When  dollars  and  cents  are  written,  a  period  or 
point  ( . )  is  placed  before  the  cents,  or  between  the  dol- 
lars and  cents.     Thus,  $7. 25  is  read  7  dollars  and  25  cents. 

71.  Since  100  cents  make  $1.00,  cents  always  occupy 
tioo  places,  and  never  more  than  two. 


ADDITION.  17 

72.  If  the  number  of  cents  is  less  than  10  and  ex- 
pressed by  a  single  figure,  a  cipher  must  occupy  the  first 
place  at  the  right  of  the  point.  Thus,  3  dollars  6  cents 
are  written  $3.06  ;  1  dollar  5  cents  are  written  $1.05. 

73.  When  cents  alone  are  written,  and  their  number 
is  less  than  100,  either  write  the  word  cents  after  the 
number,  or  place  the  dollar  sign  and  the  point  before  the 
number.     Thus,  75  cents  may  be  expressed,  $.  75. 

74.  In  arranging  for  addition,  dollars  should  be  writ- 
ten under  dollars,  and  cents  under  cents,  in  such  order 
that  the  points  stand  in  a  vertical  line. 

The  sign  $,  and  the  point  ( . )  should  never  be  omitted. 

75.  Read  the  following  equations  : 


1.  $12.  -f  $8.  =  $20. 

2.  $25. +  $10.  =  $35. 

3.  $3.25  -f  $6.75  as  $10. 


4.  $.75  +  $.20  =  $.95. 

5.  $.60  +  $.40  =  $1.00. 

6.  $14.08  -f  $3.14  =  $17.22. 


76.  Express  the  following  by  proper  figures  and  signs 


1.  Nine  dollars  and  thirty  cents. 

2.  Thirty  dollars  and  ten  cents. 

3.  Eighty-four  cents. 

4.  Seventy-eight  cents. 

5.  Six  dollars  and  sixteen  cents. 


6.  7  dollars  and  26  cents. 

7.  9  dollars  and  5  cents. 

8.  19  dollars  and  7  cents. 

9.  69  cents  ;  23  cents. 
10.  10  cents  ;  6  cents. 


The  teacher  may  exercise  the  class  orally,  by  dictating  rapidly, 
but  distinctly,  similar  examples.  Thus,  Sign,  jive,  three?  The 
prompt  response  should  be,  "  Fifty -three  dollars  "  ($53).  Ques.  Sign, 
point,  seven,  four  f  Ans.  Seventy-four  cents  ($.74).  Ques.  Signt 
point,  naught,  eight  f    Ans.  Eight  cents  ($.08),  etc. 

Also,  the  converse;  thus,  Ques.  "Forty-five  dollars"  ($45)? 
Ans.  Sign,  four,  Jive.  Ques.  Fifty-six  cents  ($.56)?  Ans.  Sign, 
point,  five,  six.  Ques.  Nine  dollars  seven  cents  ($9.07)?  Ans.  Sign, 
nine,  point,  naught,  seven  ;  etc. 


18 


ADDITION. 

7.  Copy  and  add, 

(1.)                (2.) 

(3.) 

(4.) 

$3.04            $24.12 

$105. 

$200.35 

2.21                3.06 

32.14 

46.41 

.53              12. 

.73 

1.02 

5.  What  is  the  sum  of  $.25,  $3.31,  $14.02,  and  $21. 

6.  What  is  the  sum  of  ten  dollars  and  twenty  cents, 
four  dollars  and  fifteen  cents,  forty-three  cents,  and  thir- 
teen dollars  ? 

7.  Bought  a  horse  for  $154,  aud  sold  him  for  $35.75 
more  than  he  cost.     For  how  much  did  I  sell  him  ? 

8.  A  lady  paid  12  dollars  for  a  scarf,  3  dollars  and  25 
cents  for  a  fan,  two  dollars  for  a  pair  of  gloves,  and  42 
cents  for  a  collar.     How  much  did  she  pay  for  all  ? 

It  is  not  the  design  to  teach  here,  the  principles  and  reductions  of 
Decimal  Currency,  fully  taught  in  another  place,  but  to  give  a  few 
hints  and  illustrations,  to  enable  the  teacher,  by  oral  instruction 
and  simple  written  exercises,  to  make  the  pupil  familiar  with  the 
vse  of  decimal  currency  in  common  business  matters. 


ORAL    EXERCISES. 


78.  1.  How  many  are  7  and  6 
25  and  6  ?    31  and  6  ?    42  and  6 

2.  How  many  are  9  and  7  ? 
Add 

3.  By  7's  from    1  to  71.        9. 

4.  By  7's  from    3  to  87.       10. 

5.  By  8's  from    0  to  96.       11. 

6.  By  8's  from    6  to  102.     12. 

7.  By  9's  from    2  to  92.       13. 

8.  By  9's  from  10  to  109.     14. 


?   13  and  6?   19  and  6? 

? 

16  and  7  ?     23  and  7  ? 


BylO'sfrom  0 
By  10's  from  13 
Byll'sfrom  1 
By  11 's  from  4 
Byl2'sfrom  0 
Byl2'sfrom   3 


to  120. 
to  153. 
to  100. 
to  92. 
to  144. 
to  135. 


ADDITION". 


19 


20.  13,    5,    6,  10,  and   3. 

21.  12,  10,    2,    0,  and    9. 

22.  27,    3,  10,    8,  and    7. 

23.  36,  12,    7,    4,  and  10. 

24.  11,  12,  10,    9,  and    8, 


Add  rapidly  the  following 

15.  4,  6,    5,  3,  and    7. 

16.  6,  4,    8,  2,  and    5. 

17.  10,  9,    5,  3,  and    6. 

18.  -7,  3,  10,  9,  and    8. 

19.  14,  5,    3,  6,  and  10. 

Add  by  repeating  tlie  numbers, 

25.  2,  3,  4,  2,  3,  4,  2,  3,  4,  till  the  sum  =  63. 

26.  3,  4,  5,  3,  4,  5,  3,  4,  5,  till  the  sum  =  84. 

27.  2,  4,  6,  2,  4,  6,  2,  4,  6,  till  the  sum  =  96. 

Add  alternately, 

28.  5,  6,  5,  6,  5,  6,  5,  6,  till  the  sum  =  88. 

29.  6,  4,  6,  4,  6,  4,  6,  4,  6,  4,  till  the  sum  =  100. 

30.  7,  5,  7,  5,  7,  5,  7,  5,  7,  5,  till  the  sum  =  120. 

31.  8,  9,  8,  9,  8,  9,  8,  9,  8,  9,  till  the  sum  =  119. 

32.  What  is  the  sum  of  46  and  27  ? 

Analysis. — 46  is  4  tens  and  6  units,  and  27  is  2  tens  and  7  units  ; 
4  tens  and  2  tens,  are  6  tens,  and  6  units  and  7  units  are  f8  units, 
or  1  ten  and  3  units,  which  added  to  6  tens  make  7  tens  and  3  units, 
or  73. 

44  +  37=? 
72  +  25=? 
63  +  54=? 

42.  James  earned  44  cents  one  day,  and  52  cents  the 
next.     How  many  cents  did  he  earn  in  both  days  ? 

43.  Bought  a  pound  of  coffee  for  35  cents,  a  pound  of 
butter  for  28  cents,  and  a  pound  of  sugar  for  12  cents. 
What  was  the  cost  of  the  whole  ? 

44.  A  lady  bought  a  silk  dress  for  $28,  a  shawl  for  $16, 
and  had  $14  left.     How  much  money  had  she  at  first  ? 


33. 

36  +  42=? 

36. 

54  +  38=? 

39. 

34. 

53  +  38=  ? 

37. 

29  +  61=? 

40. 

35. 

65  +  40=? 

38. 

38  +  37=? 

41. 

20  ADDITION-. 

WRITTEN     EXERCISES. 

79.  When  the  sum  of  the  units  of  any  order 
equals  or  exceeds  lO. 

1.  What  is  the  sum  of  467,  536,  84,  and  705  ? 

operation.  Analysis. — Arranging  the  numbers  as  before, 

4  6  7  begin  at  the  right  hand  and  add  the  column  of 

k  o  n  units  ;  thus,  5,  9, 15,  22  units,  equal  to  2  tens  and 

2  units.     Write  the  2  units  in  the  units'  place, 

and  reserve  the  2  tens  to  add  to  the  next  column. 

7  0  5  Next,  adding  the  2  tens  reserved,  to  the  column 

17  9  2  Sum.      °f  tens>  say>  2>  10,  13,  19  tens,  equal  to  1  hundred 

and  9  tens.     Write  the  9  tens  in  the  tens'  place, 

and  reserve  the  1  hundred  to  add  to  the  next  column. 

Lastly,  adding  the  1  hundred  reserved,  say  1,  8,  13, 17  hundreds, 
equal  to  1  thousand,  and  7  hundreds,  which  write  in  hundreds' and 
thousands'  places.    Hence  the  sum  is  1792. 

In  like  manner,  copy,  add,  and  prove, 


(2.) 

(3.) 

« 

(5.) 

276  miles. 

876  feet. 

$20.30 

$145.24 

3ft7     " 

94    " 

7.56 

36.60 

638     " 

142    " 

13.08 

105.08 

425     " 

507    " 

25. 

.75 

6.  Find  the  sum  of  $370.21,  $2.49,  $3.07,  and  $.94. 

7.  Find  the  sum  of  2008,  1400,  706,  300,  and  77. 

8.  If  4  loads  of  coal  weigh  respectively  1922,  1609, 
2100,  and  1873  pounds,  what  is  the  entire  weight  ? 

Eule. — I.  Write  the  numbers  so  that  figures  of  the  same 
order  stand  in  the  same  column. 

II.  Beginning  at  the  right,  add  each  column  separately, 
and  write  the  sum,  if  expressed  by  one  figure,  under  the 
column  added* 


ADDITION.  21 

III.  If  the  sum  of  any  column  consists  of  two  or  more 
figures,  write  the  unit  figure  under  that  column,  and  add 
the  remaining  figure  or  figures  to  the  next  column- 

Pkoof. — Add  each  column  in  the  reverse  direction.  If 
the  results  agree,  the  work  is  probably  correct. 

9.  The  Duke  of  Wellington's  army  at  Waterloo  con- 
sisted of  26661  infantry,  8735  cavalry,  6877  artillery,  and 
33413  allies.     What  was  the  whole  number  of  his  army  ? 

10.  Napoleon's  army  at  Waterloo  was  composed  of  in- 
fantry 48950,  cavalry  15765,  and  artillery  7732.  What 
was  the  whole  number  of  his  army  ? 

11.  Gave  $325  for  a  horse,  $275.50  for  a  carriage, 
$75.75  for  a  harness,  and  $20.62  for  a  robe.  What  was 
the  cost  of  the  whole  ? 

12.  Bought  a  pair  of  boots  for  $8.50,  an  umbrella  for 
$3.62,  a  pair  of  gloves  for  $1.25,  some  collars  for  $.75, 
and  a  hat  for  $4.     What  was  the  whole  cost  ? 

13.  A  lady  gave  $48.50  for  silk  for  a  dress,  $1?.75  for 
the  trimmings,  and  $15.62  for  making.  What  was  the 
cost  of  the  dress  ? 


(14.) 

(15.) 

(16.) 

(17.) 

(18.) 

(19.) 

$99.84 

96256 

$117.76 

98304 

1728 

$675.84 

24.96 

6016 

29.44 

6144 

864 

168.86 

6.24 

376 

7.36 

384 

108 

10.56 

1.56 

141 

1.84 

24576 

81 

1.32 

12.48 

188 

3.68 

3072 

5296 

.96 

.98 

1504 

58.88 

144 

3456 

2.64 

3.12 

752 

1.38 

49152 

432 

84.48 

22  ADDITION. 

20.  What  is  the  sum  of  5736  dollars  and  45  cents,  1000 
dollars  and  80  cents,  405  dollars  and  15  cents,  50  dollars 
and  9  cents,  and  7'9  cents  ? 

21.  Find  the  sum  of  twenty-five  hundred  dollars,  420 
dollars  and  47  cents,  $23  and  fifty  cents,  $600,  and  ten 
dollars  and  eight  cents. 

22.  1  million  400  thousand  and  50  -f-  15  hundred  +  25 
thousand  +  120  thousand  6  hundred  and  14  =  ? 

23.  Paid  $3456  for  a  house,  $426.75  for  painting  it, 
$2809. 48  for  furniture.     What  was  the  cost  of  the  whole  ? 

24.  North  America  has  an  area  of  8825537  square  miles, 
South  America  6954131  square  miles,  and  the  West  In- 
dies 93810  square  miles.  What  is  the  area  of  the  entire 
American  Continent  ? 

25.  A  man  owns  farms  valued  at  $62500,  city  lots 
worth  $10260,  a  house  worth  $21300,  and  other  property 
to  the  amount  of  $10500.  What  is  the  total  value  of  his 
property  ? 

80.*        SYNOPSIS    FOR    REVIEW. 


o 

l. 

2. 

Definitions.             « 
Pkinciples,  1  and  2. 

1. 

Addition.    2.  Sum  or  Amount. 
3.  Sign  of  Addition.     4.  Sign 
of  Equality.     5.  An  Equation. 
G.  Members  and  Terms  of  an 
Equation. 

3. 

4. 
,5. 

Addition  of 

< 
Dollars  and  Cents. 

Rule,  I,  II,  III. 
Proof. 

fl. 

2. 

3. 

4. 
,5. 

Sign  of  Dollars. 
Use  of  the  Period. 
Number  of  places  for  cents. 
Mode  of  expressing  cents. 
How  to  arrange  for  Addition, 

_-4b-.3ipE.Jfc 


ORAL      EXERCISES. 

81.  1.  If  John  is  15  years  old  and  George  is  6,  what  ia 
the  difference  in  their  ages  ? 

2.  How  many  are  16  cents  less  7  cents  ? 

3.  How  many  are  18  dollars  less  5  dollars  ? 

4.  How  many  are  14  less  6  ?     16  less  4  ?    12  less  5  ? 

5.  How  many  are  18  less  8  ?    20  less  6  ?    21  less  4  ? 

6.  Five  balls  taken  from  11  balls  leave  how  many? 

7.  Six  cents  from  20  cents  leave  how  many  ? 

8.  What  kind  of  numbers  are  10  days  and  6  pounds  ? 

9.  Can  6  miles  be  taken  from  15  acres  ?    Why  not  ? 

10.  Can  8  dollars  be  taken  from  18  dollars  ?    Why  ? 

11.  How  many  are  7  less  5  ?    17  less  5  ?    27  less  5  ? 

12.  How  many  are  9  less  6  ?    19  less  6  ?    29  less  ^  ? 

13.  What  number  added  to  8  will  make  12  ? 

14.  What  number  and  9  make  13  ?    14  ?    15  ?    16  ? 

15.  Subtract  by  2's  from  24  to  0. 
Operation.— 24,  22,  20,  18,  16,  14,  12,  10,  8,  6,  4,  2,  0. 
In  the  same  manner,  subtract 


16.  By  2's  from  25  to  1. 

17.  By  2's  from  31  to  3. 

18.  By  3's  from  30  to  0. 

19.  By  3's  from  37  to  1. 

20.  By  3's  from  40  to  4. 

21.  By  4's  from  44  to  0. 


22.  By  4's  from  41  to  1. 

23.  By  4's  from  51  to  3. 

24.  By  5's  from  60  to  0. 

25.  By  5's  from  63  to  3. 

26.  By  6's  from  66  to  0. 

27.  By  6's  from  65  to  5. 


24  SUBTRACTION. 

28.  Count  by  4's  from  2  to  58,  and  back  from  58  to  2. 

29.  Count  by  5's  from  1  to  61  and  back  to  1. 

30.  Count  by  6's  from  3  to  69  and  back  to  3. 

31.  Count  by  4's  from  5  to  53  and  back  to  5. 

32.  Count  by  6's  from  7  to  67  and  back  to  7. 

DEFINITIONS. 

82.  Subtraction  is  the  process  of  finding  the  dif- 
ference between  two  numbers. 

83.  The  Minuend  is  the  greater  of  the  two  num- 
bers. 

84.  The  Subtrahend  is  the  smaller  of  the  two 
numbers. 

85.  The  Difference  or  Remainder  is  the  re- 
sult obtained  by  subtracting. 

86.  The  Sign  of  Subtraction  is  — .  It  is  read 
minus,  and  signifies  less. 

When  placed  between  two  numbers,  it  indicates  that  the  one 
after  it  is  to  be  subtracted  from  the  one  before  it.  Thus,  12  —  7  is 
read  12  minus  7,  and  means  that  7  is  to  be  subtracted  from  12. 

87.  A  Parenthesis  (  )  is  used  to  include  within 
it  such  numbers  as  are  to  be  considered  together.  A 
Vinculum  has  the  same  signification.     Thus, 

25  —  (12  -f  7),  or  25  —  12  +  7,  signifies  that  from  25 
the  sum  of  12  and  7  is  to  be  subtracted. 

88.  Principles. — 1.  Only  like  numbers  and  units  of 
the  same  order  can  he  subtracted,  the  one  from  the  other. 

2.  The  minuend  must  be  equal  to  the  sum  of  the  subtra- 
hend and  remainder. 


SUBTRACTION.  25 

EXER  C  ISES. 

89.  To  be  given  in  the  same  manner  as  those  in  Art.  67. 


14—9  = 

;     ? 

24—  9=  ? 

21—12=? 

15-6  = 

;    ? 

11-  6=? 

18—10=  ? 

22-8  = 

:? 

.      17-12=? 

22-  8=  ? 

13-7  = 

-  9=? 

;    ? 

23—  9=? 

19-  7=? 

21- 

16+   7-10=? 

19-12  +  11=? 

26- 

-  7=  ? 

20+  9-  7=? 

22-11  +  15=? 

20- 

-12=? 

23+   5-12=? 

26-10  +  14=? 

25- 

-11=? 

24+   6-11=? 

29—  8+   6=? 

"     What  is  the  difference  between 

17  and  4  +  6?        20  and  6+  6?  12+  5  and  24  +  3? 

24and9  +  5?        35and9  +  20?  27—  8andl4  +  8? 

18and7  +  7?        28and9+9?  30— 10  and    9  +  8? 

Similar  dictation  exercises  may  be  given  by  the  teacher. 
WRITTEN    EXERCISES. 

90.  When  each  figure  of  the  subtrahend  is  not 
greater  than  the  corresponding  figure  of  the  min- 
uend. 

1.  From  798  subtract  563. 

operation.  Analysis, — Write  the  less  number  under  the 

M  m  q  o      greater,  so  that  units  of  the  same  order  stand 

in  the  same  column. 
Subtrahend    Obd  Begin  ftt  the  rightj  and  gubtract  each  order 

Remainder  23  5  of  units  separately  ;  thus,  3  units  from  8  units 
leave  5  units,  which  write  in  the  units'  place ; 
6  tens  from  9  tens  leave  3  tens,  which  write  in  the  tens'  place  ; 
5  hundreds  from  7  hundreds  leave  2  hundreds,  which  write  in  the 
hundreds'  place.    Hence  the  remainder  is  235. 

The  remainder  235  added  to  the  subtrahend  563  equals  798,  the 
minuend.     Hence  the  work  is  correct.     (Prin.  2.) 


m 


SUBTRACTION 


Copy,  subtract, 

and  prove, 

(2.) 

(3.) 

(4.) 

(5.) 

Minuend        426 

573 

784 

837 

Subtrahend    214 

321 

434 

315 

(6.) 

.     (?•) 

(8.) 

(9.) 

From  624  feet. 

795  tons. 

864  men. 

$976 

Take    211      " 

352    " 

413     " 

$525 

91.  Before  subtracting  dollars  and  cents,  the  numbers 
must  be  written  as  in  Addition,  in  such  order  that  the 
points  will  stand  in  the  same  vertical  line. 

In  like  manner,  subtract  and  prove, 
10.  $54.26  from  $68.37.    13.  1763  tons  from  3886  tons. 


11.  2714  from  5945. 

12.  $30.52  from  $81.76. 

16.  $93.64—  $52.41=  ? 

17.  $270.59—  $40.16=? 

18.  $703.42-$501.30=  ? 


14.  6245  feet  from  8569  feet. 

15.  7301  days  from  9625  days. 

19.  437615—213502=  ? 

20.  732740—  11520=  ? 

21.  242674-  32142=  ? 


Find  the  difference  between 


22.  1204  and  5379. 

23.  1320  and  1471. 

24.  8673  and  3560. 


25.  $57.46  and  $18. 00 +  $24.25. 

26.  $50.20  +  $4.01  and  $76.31. 

27.  $98.76  and  $30. 46 +  $43. 04. 

28.  From  five  thousand  seven  hundred  and  forty,  take 
3  thousand  and  30. 

29.  From  46  thousand  5  hundred  and  27,  take  12 
thousand  3  hundred  and  fourteen. 

20.  Two  men  bought  a  piece  of  property  for  $358.50. 
One  paid  $146.30 ;  how  much  did  the  other  pay  ? 


SUBTRACTION.  27 

31.  A  house  and  lot  sold  for  $7856,  which  was  one 
thousand  one  hundred  and  ten  dollars  more  than  the  cost. 
What  was  the  cost  ? 

32.  A  certain  city  has  a  population  of  246857,  which 
is  25324  more  than  it  had  last  year.  What  was  its 
population  last  year  ? 

ORAL    EXERCISES. 

92.  1.  A  man  having  $20,  paid  $7  for  a  hat,  and  $8 
for  a  vest.     How  many  dollars  had  he  left  ? 

Analysis. — The  difference  between  $20,  and  the  sum  of  $7  and 
$8,  which  is  $5. 

2.  A  boy  had  25  cents,  and  gave  15  cents  for  a  slate 
and  10  cents  for  some  paper.  How  many  cents  had  he  left  ? 

3.  Ella  having  16  cents,  Jane  gave  her  9  more,  and 
James  gave  her  enough  to  make  her  number  36.  How 
many  did  James  give  her  ? 

4.  Subtract  by  7's  from  63  to  0. 


5.  By  7's  from  80  to  3. 

6.  By  8's  from  64  to  0. 

7.  By  8's  from  85  to  5. 

8.  By  9's  from  90  to  0. 

9.  By  9's  from  86  to  5. 


10.  By  10's  from  100  to  0. 

11.  By  ll's  from  119  to  9. 

12.  By  ll's  from  125  to  4. 

13.  By  12's  from  129  to  9. 

14.  By  12's  from  150  to  6. 


15.  Count  by  7's  from  2  to  86,  and  back  from  86  to  2. 

16.  Count  by  8's  from  4  to  100,  and  back  to  4. 

17.  Count  by  9's  from  7  to  115,  and  back  to  7. 

18.  Count  by  10's  from  16  to  136,  and  back  to  16. 

19.  Count  by  ll's  from  9  to  119,  and  back  to  9. 

20.  Count  by  12's  from  20  to  140,  and  back  to  20. 

21.  How  many  are  5  tens  less  3  tens  ?    50  —  30  ? 


28 


SUBTKACTION 


22.  How  many  are  6  tens  less  4  tens  ?     60  —  40  ? 

23.  From  6  tens  5  units  subtract  4  tens  3  units. 

24.  From  8  tens  7  units  subtract  5  tens  6  units. 

25.  From  a  cask  containing  52  gallons,  27  gallons  were 
drawn  out.     How  many  gallons  remained  ? 

Analysis. — The  difference  between  52  gallons  and  27  gallons 
27  is  2  tens  and  7  units.  2  tens  or  20  from  52  leaves  32,  and  7  from 
32  leaves  25.     Hence  25  gallons  remained  in  the  cask. 

26.  From  a  piece  of  cloth  containing  46  yards,  24  yards 
were  cut.     How  many  yards  were  left  ? 

27.  A  man  bought  a  watch  for  $40,  and  a  chain  foi 
$15,  and  sold  both  for  $63.    How  much  did  he  gain  ? 

28.  How  many  are  6  and  40,  less  5  and  20  ? 

29.  How  many  are  7  and  30,  taken  from  5  and  50  ? 

30.  Eighteen  plus  12  equals  40  minus  how  many  ? 

31.  Twenty-two  plus  15,  equals  how  many  plus  10  ? 

32.  William  having  75  cents,  gave  25  cents  for  a  book 
and  20  cents  for  a  slate.     How  many  cents  had  he  left  ? 

33.  A  farmer  sold  a  horse  for  $96,  which  was  $23  more 
than  the  horse  cost.    What  did  he  cost  ? 

Find  the  omitted  term  in  the  following  equations  : 


34. 

12+  8—  6=? 

43. 

54_12=?  +12 

35. 

46  +  12  —  14=? 

44. 

17  +  23=56—? 

36. 

57—13+   8=? 

45. 

18  +  25=23+  ? 

37. 

60— (24  +  6)=? 

46. 

64-48=30-  ? 

38. 

28+   6=40—? 

47. 

75-30=?  +15 

39. 

42—12=18+  ? 

48. 

16  +  38=60—? 

40. 

30  +  25=?  +40 

49. 

43+  ?  =27  +  28 

41. 

27—11=19—? 

50. 

80—  ?  =100—40 

42. 

36  +  16=60—? 

51. 

22  +  54=64+  ? 

SUBTKACTION 


29 


8  14  13 

Minuend 

953 

SubtraheDd 

674 

Remainder 

279 

WRITTEN    EXERCISES. 

93.  When  any  figure  of  the  subtrahend  is  greater 
than  the  corresponding  figure  of  the  minuend. 

1.  From  953  subtract  674. 

Analysis. — Write  the  numbers  as  before 
(90),  and  subtract  each  order  of  units  sepa- 
rately. 

Since  4  units  cannot  be  subtracted  from  3 
units,  increase  the  3  units  by  a  unit  from  the 
next  higher  order,  or  10  units,  making  13  units. 
4  units  from  13  units  leave  9  units,  which  write  in  the  units'  place. 
Since  1  of  the  tens  was  united  with  the  units,  there  are  4  tens  left. 
As  7  tens  cannot  be  subtracted  from  4  tens,  increase  the  4  tens  by 
a  unit  from  the  next  higher  order,  or  10  tens,  making  14  tens. 
7  tens  from  14  tens  leave  7  tens,  which  write  in  the  tens'  place. 

Since  1  of  the  hundreds  was  united  with  the  tens,  there  are  8 
hundreds  left.  6  hundreds  from  8  hundreds  leave  2  hundreds, 
which  write  in  the  hundreds'  place.    Hence  the  remainder  is  279. 

In  like  manner,  solve  and  prove  the  following  : 

(2.)  (3.)  (4.)  (5.) 

From       3273  6345  5702  7465 

subtract  1425  2462  4384  3270 


(6.) 

(?•) 

(8.) 

From 

42670  miles. 

51062  acres. 

246700  feet. 

Take 

14384     " 

24300     " 

18030    " 

When  one  of  the  given  numbers  contains  cents,  and  the 
other  does  not,  fill  the  vacant  places  with  two  ciphers. 

(9.)  (10.)  (11.)  (12.) 

From      $325.17         $279.00         $105.08         $7.00 
Take  84.36  183.42  67.00  .84 


30  SUBTRACTION". 

Eule. — I.  Write  the  subtrahend  under  the  minuend, 
placing  units  of  the  same  order  in  the  same  column. 

II.  Begin  at  the  right,  and  subtract  the  units  of  each 
order  of  the  subtrahend  from  the  units  of  the  correspond- 
ing order  of  the  minuend,  and  write  the  result  beneath. 

III.  If  the  units  of  any  order  of  the  subtrahend  are 
greater  than  the  units  of  the  corresponding  order  of  the 
minuend,  increase  the  latter  by  10,  and  subtract ;  then 
diminish  by  1  the  units  of  the  next  higher  order  in  the 
minuend,  and  proceed  as  before. 

Proof. — Add  the  remainder  to  the  subtrahend,  and  if 
the  sum  is  equal  to  the  minuend,  the  worlc  is  correct. 

Instead  of  diminishing  by  1  the  units  of  the  next  higher  order  in 
the  minuend,  we  may  increase  by  1  the  units  of  the  next  higher 
order  in  the  subtrahend. 


Subtract 

13.  20762  from  53120. 

14.  $73.16  from  $138. 

15.  $247  from  $382.28. 


From 

16.  $430.09,  take  $272.46. 

17.  15200  rods,  take  6472  rods. 

18.  120764  tons,  take  75028  tons. 


How  many  years  from  the  date  of  each  of  the  following 
events  to  the  present  year  ? 

19.  Figures  were  used  by  the  Arabs  in  the  year  890. 

20.  Decimal  fractions  were  invented  in  1464. 

21.  Printing  was  invented  in  1441. 

22.  The  telescope  was  invented  by  Galileo  in  1610. 

23.  The  electric  telegraph  was  first  used  in  the  United 
States  in  1844. 

24.  The  first  passage  of  the  Atlantic  Ocean  by  steam 
was  in  1839. 


REVIEW.  31 

What  is  the  difference  between 

25.  34726  and  47062  ?  28.  7620  and  12420  ? 

26.  57600  and  20012  ?  29.  $4027  and  $703.41  ? 

27.  70361  and  1005  ?  30.  $1076  and  $2340.50  ? 

31.  2762  +  10341  and  45701 +  1200? 

32.  3000  +  42301  and  720  +  1684  +  7342? 

33.  A  merchant  bought  a  quantity  of  goods  for  $1248.65, 
and  sold  them  for  $1540.     How  much  did  he  gain  ? 

34.  Sold  a  horse  for  $250.75,  which  was  $28  more  than 
he  cost.     How  much  did  he  cost  ? 

35.  A  man  having  $15740.80,  gave  $5085  for  a  store, 
and  $7640. 75  for  goods.     How  much  money  had  he  left  ? 

36.  If  a  piece  of  property  bought  for  $7086.86  is  sold 
at  a  loss  of  $1562.09,  for  how  much  is  it  sold  ? 

Find  the  second  member  of  the  following  equations  : 

37.  12346  +  840  +  1046—3846=  ? 

38.  $210  +  $809.76— ($15.21 +  $308. 76)=  ? 

39.  $600.09  —  $276.25  +  $5682 — $654  =  ? 

40.  $1032.07  +  $68.05-f  $.98— $1000=  ? 

41.  476281-12672-8720  +  20000=  ? 

REVIEW. 

OKAL    EXAMPLES. 

94.  1.  The  sum  of  two  numbers  is  46,  and  one  of 
them  is  18 ;  what  is  the  other  ? 

2.  The  difference  of  two  numbers  is  16,  and  the  greater 
is  32  ;  what  is  the  less  ? 

3.  The  difference  of  two  numbers  is  24,  and  the  less 
is  26  \  what  is  the  greater  ? 


32  SUBTE  ACTION. 

4.  A  boy  having  28  peaches  gave  8  to  his  brother,  7  to 
his  sister,  and  lost  4;  how  many  had  he  left? 

5.  If  a  lady  buy  some  thread  for  10  cents,  some  needles 
for  5  cents,  and  some  ribbon  for  20  cents,  and  give  the 
clerk  50  cents,  how  much  change  should  he  return  ? 

6.  In  a  garden  are  47  fruit  trees  ;  15  of  them  are  peach 
trees,  12  plum  trees,  and  the  remainder  pear  trees.  How 
many  pear  trees  are  there  ? 

7.  A  lady  having  3  ten-dollar  bills  and  1  five-dollar 
bill,  bought  a  bonnet  for  $11,  a  pair  of  gaiters  for  $7, 
and  a  scarf  for  $3.     How  much  money  had  she  left  ? 

8.  A  man  died  at  the  age  of  64  years,  having  been 
married  36  years.     What  was  his  age  when  he  married  ? 

9.  In  a  public  school  there  are  75  pupils,  and  47  of 
them  are  girls ;  how  many  of  them  are  boys  ? 

10.  A  man  sold  25  sheep,  then  bought  12,  and  then 
had  20.     How  many  had  he  at  first  ? 

11.  A  merchant  gave  $52  for  a  box  of  goods,  and  paid 
$5  freight ;  for  how  much  must  he  sell  them  to  gain  $15  ? 

12.  A  man  gave  his  watch  and  $10  in  money  for  a  har- 
ness valued  at  $75.     How  much  did  he  get  for  his  watch  ? 

13.  A  man  having  received  $45  for  labor,  paid  $15  for 
a  coat,  $7  for  a  barrel  of  flour,  and  $6  for  a  ton  of  coal. 
How  much  had  he  left  ? 

14.  A  man  bought  a  vest  for  $7,  a  pair  of  pants  for 
$12,  and  three  shirts  for  $9,  and  gave  in  payment  3  ten- 
dollar  bills.     How  much  change  should  he  receive  ? 

Find  the  required  term  in  the  following  equations  : 


15.     42— (10  + 12)=? 

18. 

36—8  +  9  +  12=? 

16.       9  +  16=30-? 

19. 

7  +  16—8=22—  ? 

17.     36—14=15+  ? 

20. 

14  +  28-16-9=  ? 

REVIEW.  33 

WRITTEN    EXAMPLES. 

95.  1.  The  subtrahend  is  260346,  and  the  remainder 
72304.     What  is  the  minuend  ? 

2.  The  difference  is  $310.62,  and  the  minuend  $1206.28. 
What  is  the  subtrahend  ? 

3.  W^hat  is  the  sum  of  4062  and  12356  increased  by 
the  difference  between  15000  and  975  ? 

4.  From  the  sum  of  23462  and  9030,  subtract  the  dif- 
ference between  34000  and  7640. 

5.  From  the  difference  between  19876  and  6032,  sub- 
tract the  difference  between  12000  and  673. 

6.  From  what  sum  must  $.62  be  taken  to  leave  a 
remainder  of  $14. 60  ? 

7.  There  were  67374  miles  of  railway  in  the  United 
States  in  1872,  and  71564  miles  in  1873.  How  much 
was  the  gain  in  one  year  ? 

8.  A  man  has  $10000.  How  much  must  he  add  to 
this,  to  be  able  to  pay  for  a  farm  worth  $13640  ? 

9.  California  contains  158933  square  miles,  and  Texas 
237321  square  miles.  How  much  larger  is  Texas  than 
California  ? 

10.  Mt.  Blanc  is  15572  feet  high,  and  Pike's  Peak 
12000  feet.     What  is  the  difference  in  their  height  ? 

11.  A  man  willed  $125000  to  his  wife  and  two  children. 
To  his  son  he  gave  $44675,  to  his  daughter  $26380,  and 
the  remainder  to  his  wife.     What  was  his  wife's  share  ? 

12.  A  merchant  of  Nashville  goes  to  New  Orleans  with 
$21600.  He  invests  $7638.50  in  groceries,  $3210.65  in 
crockery,  $1245.18  in  woodenware,  and  the  remainder  in 
hardware.     How  much  does  he  invest  in  hardware  ? 


34  SUBTRACTION. 

13.  The  population  of  London  in  1870  was  3250000 ; 
of  New  York,  944292  ;  and  of  Brooklyn,  396099.  How 
much  greater  was  the  population  of  London  than  of  New 
York  and  Brooklyn  ? 

14.  A  man  owns  property  valued  at  $75860,  of  which 
$45640  is  invested  in  real  estate,  $25175.75  in  personal 
property,  and  the  remainder  he  has  in  bank.  How  much 
has  he  in  hank  ? 

15.  Three  persons  bought  a  hotel  valued  at  $42075. 
The  first  agreed  to  pay  $8375.50,  the  second  agreed  to 
pay  twice  as  much,  and  the  third  the  remainder.  How 
much  was  the  third  to  pay  ? 

16.  A  had  $725.40,  B  had  $180.36  more  than  A,  and 
C  had  as  much  as  A  and  B  together  minus  $214.  How 
much  had  0  ? 

17.  376  +  1684  +  573— (931  +  1000)=? 

18.  $27.62 +  $30.50— $14.00— $7.62=? 

19.  17300  +  6840-(5800  +  1386)=25300—  ? 

20.  (48036-7690)-(3600  +  1873)=18321+  ? 

96.  SYNOPSIS    FOR    REVIEW. 

1.  Subtraction.     2.  Difference,  or 
Remainder.  3.    Minuend. 


fl. 

Definitions.             + 

4.    Subtrahend.       5.    Sign  of 

ft 

Subtraction.    C.  A  Parenthesis, 

o 

L 

or  Vinculum. 

E-i 

2. 

Principles,  1  and  2. 

-** 

rl. 

How  the  numbers  should  be 

«  , 

3. 

Subtraction  op 

written. 

H 
PQ 
0 

Dollars  and  Cents. 

2. 

If  one  number  contains  cents, 

L 

and  the  other  does  not. 

m 

4. 

Rule,  I,  II,  III. 

* 

.5. 

Proof. 

ORAL    EXERCISES. 

97.  1.  If  a  man  earns  $3  a  day,  how  many  times  $3 
does  he  earn  in  4  days  ?    $3  -f-  $3  -f  $3  +  $3  are  how  many  ? 

2.  There  are  7  days  in  1  week.     How  many  days  are 
there  in  3  weeks  ?    How  many  are  three  7's,  or  3  times  7  ? 

3.  There  are  4  pecks  in  1  bushel.     How  many  pecks 
in  4  bushels  ?    Four  4's,  or  4  times  4  are  how  many  ? 

4.  Is  the  result  the  same  whether  we  say  4  times  6, 
or  6  times  4  ? 

5.  What  is  the  difference  between  six  5's  and  five  6's  ? 

6.  How  many  are  three  8's  ?    Eight  3's  ? 

7.  Add  by  2's  from  0  to  24. 

8.  Multiply  from  0  times  2,  to  12  times  2. 

Operation. — 0  times  2  is  0,  once  2  is  2,  twice  2  are  4,  3  times  2 
are  6,  4  times  2  are  8,  5  times  2  are  10,  and  so  on. 

9.  Subtract  by  2's  back  from  24  to  0. 

10.  Multiply  back  from  12  times  2  to  0  times  2. 

Operation. — 12  times  2  are  24,  11  times  2  are  22,  10  times  2  are 
20,  9  times  2  are  18,  8  times  2  are  16,  and  so  on. 

11.  Multiply  from  0  times  3  to  12  times  3,  and  back. 

12.  Multiply  from  0  times  4  to  12  times  4,  and  reverse. 

13.  Multiply  from  0  times  5  to  12  times  5,  and  reverse. 

14.  Multiply  from  0  times  6  to  12  times  6,  and  reverse. 


36 


MULTIPLICATION, 


Multiplication 

Table. 

1 

2|  3 

4 

5 

6  |  1 

8 

9 

10 

ii !  12 ! 

2 

4 

6 

8 
12 

10 
15 

12 

18 

14 
21 

16 
24 

18 

20 

22 

24 

3 

6 

9' 

27 

30 

33 

36 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

6 

12 

18 

24 

30 

36 

42 

48 

54 

60 

66 

72 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

77 

84 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

108 

10 
11 

20 
22 

30 
33 

36 

40 
44 

50 
55 

60 

70 

80 

90 

100 

110 

120 

66 

77 

88 

99 

110 

121 

132 

12 

24 

48 

60 

72 

84 

96 

108 

120 

132 

1441 

DEFINITIONS. 

98.  Multiplication  is  the  process  of  taking  one  of 
two  numbers  as  many  times  as  there  are  units  in  the  other. 
Or,  it  is  a  short  method  of  adding  equal  numbers. 

99.  The  Multiplicand  is  the  number  to  be  mul- 
tiplied. 

100.  The  Multiplier  is  the  number  by  which  to 
multiply.  It  shows  how  many  times  the  multiplicand  is 
to  be  taken. 

101.  The  Product  is  the  result  obtained  by  the  mul- 
tiplication. 

The  multiplicand  and  multiplier  are  called  the  factors  of  the 
product 


MULTIPLICATION.  37 

102.  The  Sign  of  Multiplication  is  x .    It  is 

read  times,  or  multiplied  by. 

When  placed  between  two  numbers,  it  shows  that  they  are  to  be 
multiplied  together.  Thus,  9  x  7  is  read  9  multiplied  by  7,  or  7  times  9. 

Since  changing  the  order  of  the  factors  does  not  change  the  re- 
sult, 9x7  may  be  read,  7  times  9,  or  9  times  7. 

103.  Principles. — 1.  The  multiplier  is  always  re- 
garded as  an  abstract  number. 

2.  The  multiplicand  and  product  are  like  numbers,  and 
may  be  either  concrete  or  abstract. 

In  examples  containing  concrete  numbers,  the  concrete  number  is 
the  true  multiplicand,  but  when  it  is  the  smaller,  it  is  often,  for 
convenience,  used  abstractly  as  the  multiplier. 

ORAL    EXERCISES. 

104.  8x4=?  10x0=?  llx   8=? 

7x7=?  9x5=?  9x12=? 

6x9=?  7xll=?  11x10=? 

0x5=?  12  x   6=?  10x12=? 

10x8=?  9x   7=?  9xll=? 


8x   8  +  10=?         10x10-14=?  15  x   2  +  15=? 

9x4-10=?           7x12  +  16=?  11x11-9=? 

12  x   6  +  15=?           8x   0-  7=?  12  x   0  +  25=? 

10x12-25=?           0xl2x   8=?  10x12-16=? 

8x11-12=?           1x12+  8=?  12x11-12=? 

9x   9  +  19=?          12x10-30=?  12x12+   6=  ? 

1.  At  7  cents  each,  what  is  the  cost  of  5  pencils  ? 

Analysis.— Since  1  pencil  costs  7  cents,  5  pencils  will  cost  5 
times  7  cents,  or  35  cents. 


38  MULTIPLICATION. 

2.  What  is  the  cost  of  4  tons  of  coal,  at  $8  a  ton  ? 

3.  "What  is  the  cost  of  5  hats,  at  $5  a  piece  ? 

4.  At  9  cents  each,  what  will  3  melons  cost  ? 

5.  What  will  5  yards  of  gimp  cost,  at  11  cents  a  yard  ? 

6.  12  inches  make  a  foot.  How  many  inches  in  4  feet  ? 

7.  At  $4  a  cord,  what  will  9  cords  of  wood  cost  ? 

8.  Multiply  from  0  times  6  to  12  times  6,  and  reverse. 

9.  Multiply  from  0  times  7  to  12  times  7,  and  reverse. 

10.  Multiply  from  0  times  8  to  12  times  8,  and  reverse. 

11.  Multiply  from  0  times  9  to  12  times  9,  and  reverse. 

12.  What  cost  6  pairs  of  boots  at  $8  a  pair  ?    At  $9  ? 

13.  At  8  cents  each,  what  cost  9  books  ?    10  books  ? 

14.  What  cost  9  barrels  of  flour  at  $9  a  barrel  ?  At  $10  ? 

15.  7  days  make  a  week.     How  many  days  in  7  weeks  ? 

16.  If  a  man  earn  $12  in  1  week,  how  much  will  he 
earn  in  8  weeks  ?    In  9  weeks  ? 

17.  Multiply  from  0  times  10  to  12  times  10,  and  reverse. 

18.  Multiply  from  0  times  11  to  12  times  11,  and  reverse. 

19.  Multiply  from  0  times  12  to  12  times  12,  and  reverse. 

20.  At  12  cents  a  yard,  what  cost  9  yards  of  calico  ? 

21.  What  cost  10  pounds  of  ginger,  at  11  cents  a  pound? 

22.  At  $11  a  hundred,  what  will  11  hundred.posts  cost  ? 

23.  How  many  bushels  of  grain  can  be  put  in  8  bins, 
each  containing  12  bushels  ? 

24.  How  many  are  8  times  $4,  minus  $7  ? 

25.  How  many  are  7  times  9  pounds,  plus  10  pounds  ? 

26.  How  many  are  6  times  12  rods,  less  20  rods  ? 

27.  James  gave  5   cents  each  for  6  oranges.     How 
much  change  should  he  receive  from  50  cents  ? 

28.  How  much  more  than  $35  will  7  tons  of  coal  cost, 
at  $6  a  ton  ? 


MULTIPLICATION.  39 

WRITTEN    EX  EM  CIS  ES. 

105.  When  the  multiplier  consists  of  but  one 
order  of  units. 

I.  How  many  are  4  times  73  ? 

1st  operation.       Analysis.— To  obtain  the  result  by  Addition. 
7  3  First  find  the  sum  of  four  3's,  or  4  times  3  units, 

n  g        which  is  12  units,  equal  to  1  ten  and  2  units. 
_,  ~         Write  the  2  units  in  the  units'  place,  and  reserve 

the  1  ten  to  be  added  to  the  sum  of  the  tens. 
73  Next,  the  sum  of  four  7's,  or  4  times  7  tens,  is 

Sum      2  9  2        28  tens>  and  38  tens  Plus  1  ten  ^served  are  29 
tens,  or  2  hundreds  and  9  tens,  which  write  in  the 
hundreds'  and  tens'  place.     Hence  the  sum  is  292. 

2d  operation.  Analysis. — In  this  operation,  the  multi- 
Multiplicand  7  3  plicand  73  is  written  but  once  ;  and  as  it  is  to 
Mult"  r  4-       ^e  taken  4  times,  write  the  multiplier  4  under 

it,  and  commence  at  the  right  to  multiply. 

Product      2  9  2        4  times  3  units  are  12  units,  or  1  ten  and  2 
units.     Write  the  2  units  in  units'  place  and 
reserve  the  1  ten  to  add  to  the  product  of  the  tens. 

Next,  4  times  7  tens  are  28  tens,  and  28  tens  plus  1  ten  reserved 
are  29  tens,  or  2  hundreds  and  9  tens,  which  write  in  the  hundreds' 
and  tens'  places.  Hence  the  product  is  292,  equal  to  the  sum  in  the 
first  operation. 

Solve  by  both  methods, 

2.  3  times  84.       4.    5  times  234. 

3.  4  times  135.      5.    6  times  352. 
S\  Multiply  4621  by  4  ;    by  5  ; 
9.  Multiply  3062  by  6  ;    by  7  ; 

What  is  the  product 

10.  Of  $5642  by  6?    by  5  ?    by  7?    by  9? 

II.  Of  20372  feet  by  7  ?    by  9?    by  5?    by  6? 


L 

6.    4  times  $204. 

I. 

7.    5  times  $425. 

by  6;     by  7. 

by 

8;     by  9. 

40  MULTIPLICATION. 

12.  What  cost  527  barrels  of  flour,  at  $9  a  barrel  ? 
Although  $9  is  the  true  multiplicand,  for  convenience,  we  may 

use  9  for  the  multiplier,  and  527  as  the  multiplicand  (103,  Note), 
but  the  product  is  dollars,  since  the  true  multiplicand  is  dollars. 
This  is  obvious,  since  527  barrels  at  $1  a  barrel,  would  cost  $527, 
and  at  $9  a  barrel,  9  times  $527,  etc. 

13.  What  cost  326  tons  of  coal,  at  $6  a  ton  ? 

14.  What  cost  1238  cords  of  wood,  at  $5  a  cord  ? 

15.  What  cost  752  pounds  of  nails,  at  7  cents  a  pound? 
Operation.— 7  cents  x  752  =  5264  cents  —  $52.64. 

When  either  factor  contains  cents,  the  product  is  cents, 
and  may  be  changed  to  dollars  and  cents  by  putting  the 
point  (.)  two  places  from  the  right,  and  prefixing  the 
sign  ($). 

(16.)  (17.)  (18.)  (19.) 

Multiply     $43.72  $136.04  87  cents.         $2.06 

By  8  7       9  6 

Product    $349.76  $952.28        $7.83  $12.36 

20.  At  6  cents  a  pound,  what  cost  675  pounds  of  rice  ? 

21.  At  $4. 37  a  yard,  what  is  the  cost  of  7  yards  of  cloth  ? 

22.  At  $124.50  an  acre,  what  will  5  acres  of  land  cost  ? 

23.  What  is  the  cost  of  8  building  lots,  at  $2015  each? 

ORAE   exercises. 

106.  1.  9  times  $12  are  $108.  Which  number  is  the 
Multiplicand  ?    The  Multiplier  ?    The  Product  ? 

2.  If  6  men  can  build  a  wall  in  7  days,  in  how  many 
days  can  1  man  build  it  ? 

Analysis. — It  will  take  1  man  6  times  as  many  days  as  it  will 
6  men,  to  build  the  wall ;  and  6  times  7  days  are  42  days.  Hence 
it  will  take  1  man  42  days. 


MULTIPLICATION.  41 

3.  If  7  men  can  do  a  piece  of  work  in  10  days,  how 
many  days  will  it  take  1  man  to  do  the  same  work  ? 

4.  How  many  horses  will  consume  as  many  bushels  of 
oats  in  1  day  as  7  horses  will  consume  in  5  days  ? 

5.  If  3  barrels  of  flour  last  9  persons  4  months,  how 
long  will  the  same  quantity  of  flour  last  1  person  ? 

6.  If  a  man  earns  $18  a  week,  and  spends  $9  for  board 
and  other  expenses,  how  much  will  he  save  in  8  weeks  ? 

7.  If  Henry  earn  $5  a  week,  and  James  $4,  how  much 
will  both  earn  in  7  weeks  ? 

8.  What  is  the  difference  in  the  cost  of  6  yards  of  rib- 
bon at  9  cents  a  yard,  and  of  6  yards  at  11  cents  a  yard  ? 

9.  What  will  be  the  cost  of  6  cows  at  $26  each  ? 

Analysis. — Six  cows  will  cost  6  times  $26.  6  times  6  units 
are  36  units,  or  3  tens  and  6  units,  and  6  times  2  tens  are  12  tens, 
which  plus  3  tens  and  6  units,  are  15  tens  and  6  units,  or  156. 
Hence  6  cows  will  cost  $156. 

10.  What  cost  7  pounds  of  figs,  at  23  cents  a  pound  ? 

11.  What  cost  8  pounds  of  coffee,  at  42  cents  a  pound  ? 

12.  At  $36  a  ton,  what  will  6  tons  of  guano  cost  ? 

13.  At  $18  a  barrel,  what  will  9  barrels  of  pork  cost  ? 

14.  At  $5  a  barrel,  what  are  33  barrels  of  apples  worth  ? 

15.  At  $7  a  week,  what  is  the  cost  of  21  weeks  board  ? 

16.  What  cost  20  pounds  of  beef,  at  12  cents  a  pound? 

17.  Two  men  start  from  the  same  place,  and  travel  in 
opposite  directions,  one  at  the  rate  of  6  miles  an  hour, 
the  other,  of  8  miles  an  hour.  How  far  apart  will  they  be 
at  the  end  of  6  hours  ?     8  hours  ?     9  hours ? 

18.  A  woman  sold  a  grocer  5  pounds  of  butter  at  30 
cents  a  pound,  and  received  in  payment  12  pounds  of 
sugar  at  9  cents  a  pound.     How  much  was  still  due  her  ? 


42 


MULTIPLICATION 


Find  the  second  member  of  the  following  equations 


19.  20  +  12-3x6=? 

20.  16-7  +  4x0=? 

21.  7x12-6x11=? 

22.  60-(0X12)  +  15=? 

23.  20x3  + (40-7x5)=? 


24.  3x0  +  4x7=? 

25.  (55-7)-20^8=  ? 

26.  (7  +  5)  — (6  +  4)=? 

27.  14x0  +  45-15=? 

28.  100-12x7  +  20-4=? 


WRITTEN    EXERCISES. 

107.  When   the    multiplier   consists    of  two    or 
more  orders  of  units. 

1.  Multiply  678  by  46. 


Multiplicand 
Multiplier 

1st  Partial  Prod. 
2d  Partial  Prod. 


4068  =  678  x  6 
2712     =678x40 


Entire  Prod.  31188  =  678x46 


opeeation.  Analysis.— Write  the 

6  7  8  numbers  as  before. 

4  q  Since  46  is  composed 

of  6  units  and  4  tens,  46 
times  any  number  is 
equal  to  6  times  the  num- 
ber, plus  4  tens,  or  40 
times  the  number. 
6  times  678  is  4068,  the 
first  partial  product.  4  tens  times  8  units  are  32  tens,  or  3  hun- 
dreds and  2  tens.  Write  the  2  tens  in  the  tens'  place,  in  the  second 
partial  product,  and  reserve  the  3  hundreds  to  add  to  the  product 
of  hundreds. 

4  tens  times  7  tens  are  28  hundreds,  and  28  hundreds  plus  3  hun- 
dreds reserved,  are  31  hundreds,  or  3  thousands  and  1  hundred. 
Write  the  1  hundred  in  the  hundreds'  place  in  the  second  partial 
product,  and  reserve  the  3  thousands  to  add  to  the  product  of 
thousands. 

4  tens  times  6  hundreds  are  24  thousands,  and  24  thousands  plus 
3  thousands  reserved,  are  27  thousands,  or  2  tens  of  thousands  and 
7  thousands,  which  write  in  the  second  partial  product.  The  sum 
of  the  partial  products  is  the  entire  product  31188. 

*  The  operations  of  multiplication  and  division,  indicated  by  signs,  must  be 
performed  before  those  of  addition  and  bubtractiou,  unless  otherwise  Indicated 
by  a  parenthesis  or  vinculum. 


MULTIPLICATION.  43 

In  like  manner,  multiply 


2.  473  by  27. 

5.  $36.45  by  34; 

by  47. 

3.  738  by  35. 

6.  $70.65  by  55; 

by  64. 

4.  609  by  56. 

7.  $29.07  by  76  ; 

by  82. 

Rule. — I.  Write  the  multiplier  under  the  multiplicand, 
so  that  units  of  the  same  order  stand  in  the  same  column. 

When  the  multiplier  consists  of  one  figure. 

II.  Begin  at  the  right  and  multiply  the  units  of  each 
order  of  the  multiplicand  by  the  multiplier.  Write  in  the 
product  the  units  of  each  result,  and  reserve  the  tens  to 
add  to  the  next  result. 

When  the  multiplier  consists  of  more  than  one  figure. 

III.  Multiply  the  multiplicand  by  the  units  of  each  order 
of  the  multiplier  successively,  beginning  at  the  right,  and 
write  the  right-hand  figure  of  each  partial  product  under 
the  order  of  the  multiplier  used. 

TJie  sum  of  the  partial  products  is  the  required  product. 

Proof. — Revieiv  the  work  carefully,  or  multiply  the 
multiplier  by  the  multiplicand  ;  if  the  results  are  the  same, 
the  work  is  probably  correct. 

When  there  are  ciphers  in  the  multiplier,  multiply  by  the  sig- 
nificant figures  only,  since  the  product  of  any  number  by  0  is  0. 

8.  Multiply  6432  by  75  ;    by  67  ;    by  136. 

9.  Multiply  23072  by  128  ;    by  243  ;    by  307. 

10.  Multiply  $420.06  by  204  ;    by  666  ;    by  408. 
What  is  the  value 

11.  Of  67  hogsheads  of  sugar,  at  $37.75  a  hogshead  ? 

12.  Of  2347  acres  of  land,  at  $136  an  acre  ? 

13.  Of  64  horses,  at  $219.75  each  ? 


44  MULTIPLICATION. 

14.  What  will  be  the  cost  of  building  a  line  of  telegraph 
274  miles  long,  at  $967  a  mile  ? 

15.  If  1049  pounds  of  seed  cotton  be  raised  from  an 
acre  of  land,  how  many  pounds  will  386  acres  produce  ? 

16.  If  a  cotton  mill  manufactures  628  yards  of  cloth 
in  a  day,  how  many  yards  can  it  make  in  297  days  ? 

What  is  the  product 


17.  Of  2572  bushels  by  94  ? 

18.  Of  $403.06  by  127  ? 

19.  Of  86072  pounds  by  208  ? 

20.  Of  316  times  $487.46  ? 

21.  Of  507  times  30975  days 

22.  Of  325  times  6408  cents  ? 

29.  Find  the  cost  of  386  railway  coaches,  at  $7034.75 
each. 

30.  What  cost  802  tubs  of  butter,  at  $27.08  each  ? 


23.  Of  370607  by  4071  ? 

24.  Of  600326  by  2645  ? 

25.  Of  730096  by  5006  ? 

26.  Of  2407068  by  3406 

27.  Of  408091  by  2407  ? 

28.  Of  73069  by  46035  ? 


31.  236x63x28=? 

32.  439x0x142=? 


33.  1927x613x802=? 

34.  4605x2034x576=? 


35.  How  many  yards  of  shirting  in  49  bales,  each  bale 
containing  26  pieces,  and  each  piece  57  yards  ? 

36.  What  is  the  cost  of  128  barrels  of  beef,  each  con- 
taining 216  pounds,  worth  13  cents  a  pound  ? 

37.  Three  schooners,  ship  239  cords  of  wood  each,  and 
a  fourth  ships  248  cords.  What  is  the  value  of  the  whole 
at  $4.25  a  cord  ? 

38.  If  it  require  108  tons  of  iron  rail  for  1  mile  of 
track,  how  many  tons  will  be  required  for  476  miles,  and 
what  will  be  its  value  at  $145  a  ton  ? 

39.  A  crop  of  cotton  was  put  up  in  472  bales,  the 
average  weight  of  which  was  588  pounds.  What  was  the 
weight  of  the  whole  crop,  and  its  value  at  18  cents  a  pound  ? 


MULTIPLICATION.  45 

108.  To  multiply  by  the  factors  of  a  number. 

The  Factors  of  a  number  are  the  numbers  which 
multiplied  together  will  produce  it.  Thus,  6  and  7  are 
factors  of  42 ;  2,  4,  and  5  are  factors  of  40. 

The  pupil  should  carefully  distinguish  between  the  factors  and 
the  parts  of  a  number.  The  factors  are  multiplied,  but  the  parts 
are  added,  to  produce  a  number.  A  factor  is  always  a  part,  but  a 
part  is  not  always  a  factor. 

Thus,  2  and  9,  3  and  6,  2,  3,  and  3,  are  factors  of  18 ;  but  the 
parts  of  18  are  9  and  9,  10  and  8,  6  and  12,  7  and  11,  etc. 

109.  Principle. — The  product  of  any  number  of  fac- 
tors will  be  the  same  in  whatever  order  they  are  multiplied* 


1.  Multiply 

468 

by  36. 

OPERATION. 

36 

=  6 

X  6,  or 

9 

X 

4,  or  12 

X 

3. 

468 

468 

468 

468 

36 

6 

9 

12 

2808 

2808 

4212 

5616 

1404 

6 

4 

3 

16848     16848     16848     16848 

It  will  be  observed  that  the  multiplicand,  multiplied  by  the 
given  multiplier,  or  by  any  set  of  factors  into  which  it  can  be  sepa- 
rated, produces  the  same  result. 


In  like  manner,  multiply 

2.  $73.04  by  48=8x6. 

3.  50076  by  72=6x4x3. 

4.  46502  by  84=7x4x3. 

5.  $206.14  by  96=4x4x6. 


6.  $780.91  by  108. 

7.  140086  by  120. 

8.  380509  by  144. 

9.  $457.52  by  240. 


4:6  MULTIPLICATION. 

Kule. — I.  Separate  the  multiplier  into  two  or  more 
factors. 

II.  Multiply  the  multiplicand  by  one  of  the  factors,  the 
resulting  product  by  another  factor,  and  so  continue  until 
all  the  factors  have  been  used. 

The  last  product  ivill  be  the  product  required. 

10.  What  will  56  acres  of  land  cost,  at  $164.50  an  acre  ? 

11.  At  28  cents  a  pound,  what  will  be  the  cost  of  24 
sacks  of  coffee,  each  containing  64  pounds  ? 

12.  What  is  the  value  of  107  pieces  of  cloth,  each  piece 
containing  42  yards,  at  $4.28  a  yard  ? 

110.  When  either  the  multiplicand  or  multi- 
plier, or  both,  have  ciphers  on  the  right. 

1.  Multiply  286  by  100. 

operation.  Analysis. — Since  removing  a  figure  one  place  to 

2  8  6  the  left,  increases  its  value  ten  times  (44),  annex- 

i  Q  o  mg  a  cipher  to  a  number  multiplies  it  by  10  ;  an- 

nexing  two  ciphers  multiplies  it  by  100,  etc.    Hence 

2  8  6  0  0  286  x  100=28600,  the  product  required. 

2.  Multiply  3240  by  600. 

operation.  Analysis.— 3240  =  324  x  10,  and  600  =  6  x  100. 

3  2  4  0  First  multiply  together  the  two  factors  324  and  6, 

qqq  and  then  multiply  their  product  1944,  by  10 x  100, 

or  by   1000,    by  annexing  three  ciphers,  which 

19  44  0  0  0        gives  1944000,  the  required  product. 

What  is  the  product 

3.  Of  372  by  10  ?     By  100  ?     By  1000  ?     By  10000  ? 

4.  Of  860  by  50?     By  400  ?     By  1500?     By  3000  ? 
Rule. — To  the  product  of  the  significant  figures,  annex 

as  many  ciphers  as  there  are  ciphers  on  the  right  of  either 
or  of  both  of  the  factors. 


REVIEW.  47 

Find 


8.  120  times  5000. 

9.  600  times  21000. 
10.     1000  times  104000. 


What  is  the  product 

5.  Of  $4. 72  by  100? 

6.  Of  $30.40  by  60  ? 

7.  Of  $1200  by  700  ? 

II.  42030090  x  3020=  ?         12.     7000600  x  50040=  ? 

13.  There  are  640  acres  in  1  square  mile.     How  many 
acres  in  150  square  miles  ?    In  200  ?    In  420  ? 

14.  The  salary  of  the  president  is  $50000  a  year.    How 
much  does  he  receive  in  8  years  ? 

REVIEW. 

ORAL    EXAMPLES. 

III.  1.  The  sum  of  8  +  12  +  16  equals  the  product  of 
9  x  what  number  ? 

2.  The  sum  of  40  —  14  and  12  +  4  equals  7  x  what 
number  ? 

3.  The  difference  between  35  +  15  and  24—10  is  equal 
to  the  product  of  what  two  factors  ?     Three  factors  ? 

4.  The  product  of  what  two  factors  is  equal  to  the 
sum  of  9,  20,  and  11  ? 

5.  The  product  of  8  times  9  is  equal  to  6  times  what 
number  ? 

6.  The  sum  of  25,  13,  8,  and  10  is  equal  to  the  pro- 
duct of  what  three  factors  ? 

7.  What  is  the  sum  of  3  times  3x4,  and  5  times  4x3? 
What  is  the  difference  ? 

8.  What  is  the  product  of  15  +  24—14  by  16—12  ? 

9.  Which  is  greater,  9  xl3— 6,  or  12  times  8—20  ? 
10.  How  much  less  is  60— 5  x8  than  16  +  14—10? 


48  MULTIPLICATION. 

11.  Charles  is  twice  as  old  as  George,  and  George  is 
12  years  old.    What  is  the  sum  of  their  ages  ? 

12.  What  is  the  cost  of  4  brooms  at  30  cents  each,  and 
6  pounds  of  sugar  at  11  cents  a  pound? 

13.  Mary  had  18  cents,  and  Belle  had  3  times  as  many 
less  9  cents.     How  many  had  both  ? 

14.  A  young  man  earned  $9  a  week,  and  spent  $5  a 
week  for  board.     How  much  did  he  save  in  12  weeks  ? 

15.  A  woman  sold  a  grocer  4  dozen  of  eggs  at  24  cents 
a  dozen,  and  received  in  payment  half  a  pound  of  tea 
worth  50  cents,  and  2  pounds  of  sugar  at  11  cents  a 
pound.    How  much  was  still  due  her  ? 

16.  A  boy  bought  a  book  for  36  cents,  a  slate  for  20 
cents,  and  a  pencil  for  4  cents.  How  much  change 
should  he  receive  from  a  1  dollar  bill  ? 

17.  A  lady  bought  9  yards  of  silk  at  $3  a  yard,  3  pairs 
of  kid  gloves  at  $2  a  pair,  4  pairs  of  hose  at  half  a  dollar 
a  pair.  She  gave  in  payment  4  ten  dollar  bills.  How 
much  change  should  she  receive  ? 

Find  the  required  term  in  the  following  equations  : 


24.  7x12—25x0=? 

25.  3x0x5  +  16=2x  ? 


18.  19—7  +  28-11=? 

19.  8x9— 16  =  7x  ? 

20.  21  +  6x7=40+  ? 

21.  10x12-9x11=? 

22.  75-5x12=35—? 

23.  44 +  19 -(50— 23)=? 

112.  By  a  little  practice,  numbers  containing  three  or 
four  figures  may  be  multiplied  mentally,  by  first  multi- 
plying the  highest  order  of  units,  and  adding  the  pro- 
duct of  each  lower  order  as  found. 


26.  28  +  12— 6  x  ?  =16 

27.  9x12  +  10=120—  ? 

28.  42—20  +  14=?  x9 

29.  8  +  55-(?  x8)=7 


REVIEW.  49 

1.  Multiply  324  by  2. 

Operation. — 2  times  3  hundreds  are  600  ;  2  times  2  tens  are  4  tens, 
or  40,  and  600  +  40  are  640  ;  2  times  4  are  8,  and  640  +  8  are  648. 

Omitting  all  but  results,  the  required  product  will  be  easily  and 
promptly  obtained  by  a  strictly  mental  process.     Thus,  600, 640,  648. 

In  like  manner,  find  the  product  of 


2. 

3 

times 

230. 

5. 

4 

times 

425. 

8. 

234  x  2. 

3. 

3 

times 

342. 

6. 

6 

times 

241. 

9. 

501  x  3. 

4. 

4 

times 

150. 

7. 

5 

times 

615. 

10. 

255  x  4. 

WRITTEN     EXAMPLES. 

113.  1.  If  I  receive  $1500  salary,  and  pay  $370  for 
board,  $281.50  for  clothing,  $112.75  for  books,  and  $196.65 
for  other  expenses  annually,  what  can  I  save  in  3  years  ? 

2.  A  merchant  bought  7  hogsheads  of  sugar  at  $46.45 
a  hogshead,  and  sold  it  for  $53.62  a  hogshead.  How 
much  did  he  gain  ? 

3.  Paid  $2709  for  388  barrels  of  flour,  and  sold  the 
same  at  $9.12  a  barrel.     How  much  was  the  gain  ? 

4.  If  a  man  have  an  income  of  $5670  a  year  and  his 
daily  expenses  average  $7.25,  how  much  can  he  save  in  a 
year  of  365  days  ? 

5.  What  number  must  be  added  to  272  x  400  to  make 
the  amount  126720  ? 

6.  What  is  the  difference  between  40706  —  308  x  5G, 
and  97  x  340— 12400? 

7.  Multiply  98  +  6  x  (37  +  50)  by  64-50  x  5  —  10. 

8.  Multiply  675 -(77 +  56)  by  (3  x  155) —  (214— 28). 

9.  A  man  owing  $15760,  gave  in  payment  5  lots  of 
land,  worth  $730  each,  5  horses,  valued  at  $236. 50  each, 
an  interest  he  had  in  a  coal  mine  worth  $2000,  and 
$1728.75  in  money.     How  much  remained  unpaid? 


50  MULTIPLICATION-. 

10.  A  farm-house  is  worth  $3246,  the  farm  is  worth 
3  times  as  much  plus  $1200,  and  the  stock  is  worth 
twice  as  much  as  the  house,  less  $1875.  What  is  the 
value  of  the  whole,  and  of  the  farm  and  stock  ? 

11.  What  is  the  difference  in  the  cost  of  48  horses  at 
$184.50  each,  and  of  130  sheep  at  $4.80  a  head  ? 

12.  Bought  150  barrels  of  flour  for  $1150,  and  finding 
25  barrels  of  it  worthless,  sold  the  remainder  at  $9  a  bar- 
rel.    Did  I  gain  or  lose,  and  how  much  ? 

Complete  the  following  equations  : 

13.  (142  +  405)  x  (1000  —  850)  -  5000  =  ? 

14.  (97  X  1000)  -  (75  x  500"-^420)  +  1500  =  ? 

15.  $73.46  —  ($.94  +  $3.02)  +  $47  x  35  =  ? 

16.  $246.08  x  104  +  ($2000  —  $240.50)  x  10  =  ? 

114.  SYNOPSIS  FOR  REVIEW. 

f  1.  Multiplication.      2.  Multiplicand. 

1.  Definitions.  i       3.  Multiplier.   4.  Product.  5.  Sign 
(^       of  Multiplication. 

2.  Principles,  1  and  2. 


3.  Rule— I,  II,  III. 

4.  Proof. 

5.  When  either  factor  contains  cents. 

f  1.  Definition  of  factors. 

6.  By  Factors.  <  2.  Principle. 

3.  Rule. 


7.  Multiplier    and 

Multipucand.     i   .    onttangk*- 


1.  When  one  or  both  have  ciphers 
on  the 

2.  Rule. 


(tang 


ORAZ      EXERCISES. 

115.  1.  How  many  4's  are  12?    Are  16  ?    Are  24? 

2.  How  many  lots,  of  5  acres  each,  in  20  acres  ? 

3.  How  many  5's  in  15  ?     In  30  ?     In  35  ?     In  50  ? 

4.  How  many  barrels,  each  holding  3  bushels,  will  be 
required  for  18  bushels  of  apples  ?    21  bushels  ? 

5.  How  many  times  can  6  yards  of  cloth  be  taken  from 
a  piece  containing  30  yards. 

6.  How  many  times  can  6  cents  be  taken  from  23  cents, 
so  as  to  have  5  cents  remaining  ?   * 

7.  Distribute  $28  equally  among  7  men.     How  many 

dollars  will  each  receive  ? 

Do  you  find  how  many  times  7  men  are  contained  in  $28,  or  do 
you  find  one  of  7  equal  parts  of  $28  ? 

8.  How  do  you  find  one  of  8  equal  parts  of  a  number  ? 
Of  9  equal  parts  ?     Of  6  equal  parts  ? 

9.  What  is  one  of  4  equal  parts  of  40  ?   Of  36  ?   Of  48  ? 

10.  What  is  one  of  6  equal  parts  of  30  ?   Of  42  ?    Of  48  ? 

11.  What  is  one  of  7  equal  parts  of  56  pounds  ? 

12.  How  many  times  8  cents  are  48  cents?    Is  the 
result  a  concrete  or  an  abstract  number  ? 

13.  What  is  one  of  8  equal  parts  of  48  cents  ?    Is  the 
result  a  concrete  or  an  abstract  number  ? 


52  DIVISION. 

DEFINITIONS. 

116.  Division  is  the  process  of  finding  how  many 
times  one  number  is  contained  in  another  of  the  same 
kind,  or  of  finding  one  of  the  equal  parts  of  a  number. 

117.  The  Dividend  is  the  number  to  be  divided. 

118.  The  Divisor  is  the  number  by  which  to  divide. 

119.  The  Quotient  is  the  result  of  the  division,  and 
shows  how  many  times  the  dividend  contains  the  divisor. 

The  division  is  said  to  be  exact  when  there  is  no  remainder. 
The  part  of  the  dividend  remaining  when  the  division  is  not  exact 
is  called  the  Remainder,  and  must  always  be  less  than  the  divisor. 

120.  The  Sign  of  Division  is  -^.     It  is  read 

divided  by. 

It  shows  that  the  number  before  it  is  to  be  divided  by  the  one 
after  it ;  thus  54  -5-  9  is  read  54  divided  by  9. 

121.  Division  is  als.o  indicated  by  placing  the  dividend 
above  the  divisor  with  a  line  between  them  ;  thus,  -^  is 
read  72  divided  by  8. 

122.  Principles. — In  finding  how  many  times  one 
number  is  contained  in  another  : 

1.  The  divisor  and  dividend  are  like  numbers,  and  the 
quotient  an  abstract  number. 

In  finding  one  of  the  equal  parts  of  a  number: 

2.  The  dividend  and  quotient  are  like  numbers,  and  the 
divisor  an  abstract  number. 

3.  The  dividend  is  equal  to  the  product  of  the  divisor 
by  the  quotient,  plus  the  remainder. 


division.  53 


OUA.E    exercises 


33.  36-=-9=  ? 

42-^-7=? 
40-^5=? 

63-^9=? 
56^8=? 
45^5=? 

64-v-8=  ? 
72-5-9=? 

84-=-  7=? 
72-5-12=  ? 
96-s-  8=? 

L  Divide  by  2, 

V  =  ?            f|  =  ? 

from  2  in  2  to  2  in  24. 

Operation. — 2  in  2,  once  ;  2  in  4,  twice  ;  2  in  6,  3  times  ;  2  in  8, 
4  times  ;  2  in  10,  5  times,  and  so  on  to  2  in  24,  12  times. 

In  the  same  manner,  divide 

2.  By  3,  from  3  in  3,  to  3  in  36. 

3.  By  4,  from  4  in  4,  to  4  in  48. 

4.  By  5,  from  5  in  5,  to  5  in  60. 

5.  By  6,  from  6  in  6,  to  6  in  72. 

6.  By  7,  from  7  in  7,  to  7  in  84. 

7.  By  8,  from  8  in  8,  to  8  in  96. 

8.  By  9,  from  9  in  9,  to  9  in  108. 

9.  By  10,  from  10  in  10,  to  10  in  120. 

The  pupil  may  reverse  the  above  ;  thus,  2  in  24, 12  times  ;  2  in  22, 

11  times  ;  2  in  20,  10  times,  and  so  on. 

Also  combine  ike  two  ;  thus,  3  in  3,  once;  3  in  6,  twice,  2  in  6, 
3  times  ;  3  in  12,  4  times,  4  in  12,  3  times ;  and  so  on  to  3  in  36, 

12  times,  12  in  36,  3  times. 

124.  Division  may  also  be  regarded  as  a  short  method 

of  performing  several  subtractions  of  a  number. 

Thus,  24-6  =  18;  18-6  =  12;  12-6  =  6;  6-6  =  0.  We 
have  performed  four  subtractions  of  6,  hence  there  are  four  6's  in 
24,  or  6  is  contained  in  24,  4  times. 


54  division. 

125.  Since  one  number  is  contained  in  another  as 
many  times  as  it  is  a  factor  of  the  other,,  division  may  be 
regarded  as  the  reverse  of  multiplication. 

In  Multiplication,  both  factors  are  given  to  find  the 
product  ;  in  Division,  one  factor  and  the  product  (an- 
swering to  the  dividend)  are  given  to  find  the  other  factor, 
which  answers  to  the  quotient. 

Thus,  6  x  4  =  24,  the  factor  6  being  taken  4  times  ;  hence  there 
are  four  6's  in  24,  or  6  is  contained  in  24,  4  times. 

126.  The  Object  of  Division  is  twofold. 

First.  To  find  hoio  many  times  one  number  is  contained 
in  another  of  the  same  hind. 

Ex.  At  5  cents  each,  how  many  pencils  can  be  bought  for  20  cents. 

Since  5  cents  taken  4  times  equals  20  cents  (5  x  4=20),  it  follows 
that  5  cents  is  contained  in  20  cents  4  times. 

Analysis. — As  many  pencils  can  be  bought  for  20  cents,  as  5  cents 
are  contained  times  in  20  cents,  which  are  4  times.    Hence,  etc. 

127.  Second.  To  separate  a  given  number  into  as 
many  equal  parts  as  there  are  units  in  another. 

Ex.  If  4  pencils  cost  20  cents,  what  is  the  cost  of  1  pencil  ? 

Since  5  cents  taken  4  times  equals  20  cents,  it  follows  that  5  cents 
is  one  of  the  four  equal  parts  of  20  cents  (5  +  5  +  5  +  5=20),  and  we 
say  one-fourth  of  20  cents  is  5  cents. 

Analysis. — Since  4  pencils  cost  20  cents,  1  pencil  costs  one-fourth 
of  20  cents,  which  are  5  cents. 

128.  The  equal  parts  into  which  a  unit  or  whole  thing 
is  divided  are  called  fractions. 

129.  The  names  of  these  equal  parts  of  a  unit  vary 
according  to  the  number  of  these  parts  ;  thus,  one-half  is 
one  of  two  equal  parts,  one-third  is  one  of  three  equal 
parts  into  which  the  whole  thing  or  number  is  divided. 


DIVISION.  55 

So  in  like  manner  we  have  fourths, fifths,  sixths,  sevenths, 
eighths,  tenths,  twelfths,  twentieths,  etc. 

130.  These  parts  are  expressed  by  writing  the  number 
denoting  the  name  of  the  parts  below  a  short  horizontal 
line  as  a  divisor,  and  the  number  of  parts  taken  or  used, 
above  the  line  as  a  dividend. 

Thus,  \,  signifies  1  divided  by  2,  and  is  read,  one-half 
|,  signifies  2  divided  by  3,  and  is  read  tivo-thirds. 
T\,  signifies  7  divided  by  12,  and  is  read,  seven-twelfths, 
etc. 

ORAL     EXERCISES. 

131.  1.  If  a  number  is  separated  into  two  equal  parts, 
what  is  each  part  called  ? 

Ans.  One-half  oi  the  number,  written  -J-. 

2.  If  $18  are  equally  divided  between  two  poor  families, 
how  much  does  each  receive  ?    What  part  of  the  whole  ? 

3.  What  is  one-half  of  12  ?     Of  16  ?    Of  20  ?     Of  24  ? 

4.  If  a  number  is  separated  into  three  equal  parts,  what 
is  each  part  called  ?       One-third  of  the  number,  -J-. 

5.  If  15  peaches  are  equally  distributed  among  3  boys, 
what  part  of  the  whole  will  each  receive  ? 

6.  What  is  one-third  oi  $15?   Of  21  days  ?   Of  30  rods'? 

7.  Divide  an  acre  of  land  into  four  equal  parts.  What 
is  one  of  the  parts  called  ?     One-fourth  of  an  acre,  £. 

8.  What  are  2  of  the  parts  called  ?  Two- fourths,  £. 
Three  of  the  parts  ?  TJiree- fourths,  J. 

9.  If  48  marbles  are  given  to  4  boys,  to  each  an  equal 
number,  what  part  of  the  whole  does  1  boy  receive  ? 
Two  boys  ?     Three  boys  ?    How  many  marbles  ? 

10.  What  is  one-fourth  of  $24  ?     Of  48  miles  ? 


56  division. 

11.  If  a  number  is  divided  into  five  equal  parts,  what  is 
each  part  called  ?  One-fifth  of  the  number,  \.  Two 
parts  ?    Two-fifths,  |. 

12.  If  $20  are  paid  for  5  barrels  of  apples,  what  part 
of  $20  is  paid  for  1  barrel  ?  For  2  barrels  ?   For  3  barrels  ? 

13.  What  is  £  of  30?     Of  $40?     Of  45  rods? 

14.  If  a  number  is  divided  into  six  equal  parts,  what  is 
each  part  called  ?  One-sixth  of  the  number,  -J-. 

15.  If  into  seven  equal  parts  ?      One-seventh,  \. 

16.  If  into  e^£  equal  parts  ?  One-eighth,  £. 

17.  If  into  nine  equal  parts  ?  One-nintJi,  -J-. 

18.  If  into  ten  equal  parts  ?  One-tenth,  -fa. 

19.  If  into  twelve  equal  parts  ?     One-twelfth,  -fa. 

20.  Find  one-half  of  2,  one-half  of  4,  one-half  of  6, 
one-half  of  8,  and  so  on  to  one-half  of  20. 

21.  Find  one-third  of  3,  one-third  of  6,  one-third  of  9, 
one-third  of  12,  and  so  on  to  one-third  of  30. 

22.  Find  \  of  4,  J  of    8,  J  of  12,  J  of  16,  to  \  of  40. 

23.  Find  \  of  5,  |  of  10,  J  of  15,  \  of  20,  to  \  of  50. 

24.  Find  £  of  6,  J  of  12,  £  of  18,  |  of  24,  to  J  of  60. 

25.  Find  |  of  7,  |  of  14,  \  of  21,  |  of  28,  to  \  of  70. 

26.  Find  \  of  8,  J  of  16,  |  of  24,  £  of  32,  to  J  of  80. 

27.  Find  £  of  9,  J  of  18,  £  of  27,  i  of  36,  to  |  of  90. 

28.  Find  3V  of  10,  ^  of  20,  ^  of  30,  to  ^  of  100. 

29.  How  do  you  find  -J ,  -J,  J,  -J,  -J-,  etc.,  of  any  number  ? 

30.  How  many  yards  of  cloth,  at  $4  a  yard,  can  be 

bought  for  $36  ? 

Analysis. — As  many  yards  as  $4  are  contained  times  in  $36, 
■which  are  9  times.    Hence  9  yards  can  be  bought  for  $36. 

31.  At  $6  a  ton,  how  many  tons  of  coal  can  be  bought 
for  $24?    For  $30?    For  $54?    For  $72? 


DIVISION.  57 

32.  If  7  cords  of  wood  cost  $42,  what  does  1  cord  cost  ? 
Analysis. — Since  7  cords  of  wood  cost  $42, 1  cord  costs  1  seventh 

of  $42,  or  $6.     Hence  1  cord  costs  $6. 

33.  A  man  sold  8  bushels  of  cranberries  for  $32.  Hew 
much  did  he  receive  a  bushel  for  them  ? 

34.  A  farmer  gathered  108  bushels  of  apples  from  9  trees. 
What  was  the  average  number  of  bushels  to  each  tree  ? 

35.  A  merchant  paid  $96  for  8  pieces  of  dress  goods. 
What  was  the  cost  of  each  piece  ? 

36.  If  a  farm  of  120  acres  is  divided  into  12  equal  lots, 
how  many  acres  does  each  lot  contain? 

WRITTEN     EXERCISES. 

132.  When  the  divisor  consists  of  but  one  order 
of  units. 

1.  Divide  875  by  7. 

Analysis. — Write  the  divisor  at  the 
operation.  left  of  tlie  dividend  witll  a  line  between 

Divisor.  Dividend.  Quotient.      them. 

7)875(125  7is  contained  in  8  hundreds,  1  hun- 
7                            dred  times,  with  a  remainder.     Write 
7"T                         the  1  hundred  at  the  right  of  the  divi- 
dend, for  the  first  figure  of  the  quotient. 
Multiply  the  divisor  7  by  the  1  hundred 
3  5                      °f  tlie  quotient,  and  write  the  product, 
ok                       7  hundreds,  under  the  hundreds  of  the 
dividend.     Subtract,  and  to  the  remain- 
der 1  hundred,  annex  the  7  tens  of  the  dividend,  making  17  tens. 

7  is  contained  in  17  tens,  2  tens  times,  with  a  remainder.  Write 
the  2  tens  in  the  quotient.  Multiply  the  divisor  7  by  the  2  tens, 
and  subtract  the  product  from  the  partial  dividend,  17  tens.  To 
the  remainder  3  tens,  annex  the  5  units  of  the  dividend,  making 
35  units. 

7  is  contained- in  35  units,  5  times,  which  write  in  the  quotient. 
Multiplying  and  subtracting  as  before,  nothing  remains.    Hence,  etc. 


58  DIVISION. 

The  solution  of  the  preceding  example  may  be  abbre- 
viated by  what  is  termed  Short  Division,  as  follows  : 

Analysis. — 7  is  contained  in  8,  once,  and 

operation,     i  remainder.     1  prefixed  to  7  makes  17.     7  is 

7)875        contained  in  17,  2  times  and  3  remainder.     3 

-i  o  k        prefixed  to  5  makes  35,  and  7  is  contained  in 
Quotient  1/3  5         *"  '  . 

35,  5  times.     Hence  the  quotient  is  125. 

133.  In  Short  Division  only  the  quotient  is  writ- 
ten,  the  operations   being  performed  mentally.     It  is 
generally  used  when  the  divisor  does  not  exceed  12. 
In  like  manner,  divide  and  analyze  the  following  : 
(2.)  (3.)  (4.)  (5.) 

6)7944  7)9464  8)8928  5)6895 

6.  Divide  92352  by  8  ;    by  6  ;    by  4. 

7.  Divide  83762  by  7  ;     79880  by  6  ;     3263  by  8. 

Analysis. — Since  8  is  not  contained  in  3 
operation,     thousands,  unite  the  3  thousands  and  2  hun- 
8)3263         dreds,  making  32  hundreds.     8  is  contained 
O  otient  40  7  4-      in  32  hundreds,   4  hundreds  times,  which 

write  in  the  hundreds'  place  in  the  quotient. 
Next,  8  is  not  contained  in  6  tens,  so  write  a  cipher  in  tens'  place 
in  the  quotient,  and  unite  the  6  tens  and  3  units.  8  is  contained  in 
63  units  7  times  and  7  units  remainder,  which  write  over  the  divi- 
sor and  add  as  a  part  of  the  quotient.  Hence  the  quotient  is  407£. 
Proof. — Multiply  the  quotient  407  by  the  divisor  8,  and  the 
product  is  3256 ;  3253  plus  the  remainder  7,  equals  the  dividend 
3263.     (Prin.  3.) 

8.  Divide  8135464  by  6  ;    by  8  ;    by  7  ;    by  5  ;    by  9. 

9.  Divide  $48.56  by  8  cents. 

Eight  cents  may  be  written  $.08  (73). 

operation.  When  the  divisor  and  dividend  are 

$.08)$48.56  like  numbers,  the  quotient  is  an  abstract 

VcTv  +•  number  (Prin.  1).    Hence  8  cents  are 

0  0  7  times.     contained  in  $48.56,  607  times. 


division.  59 

10.  Divide  $48.56  by  8. 

operation.  When  the  divisor  is  an  abstract  number,  the 

8  )  $  4  8  .  5  G      dividend  and  quotient  are  like  numbers  (Prln.  2). 
$^J7o~7     Hence  1  eighth  of  $48.56  is  $6.07. 

Solve  and  prove, 

(11.)  (12.)  (13.)  (14.) 

9)1217.62      7)16.44      $7)1644  $.07)  $6.44 

$24.18  $.92  92  times.  92  times. 

How  many  times 

15.  Are  $8  contained  in  $15096  ?  In  $58424  ?  In  $23064  ? 

16.  Is  7  contained  in  330457  ?    In  19278  ?  In  918271  ? 

17.  Is  9  contained  in  436281  ?   In  605675  ?  In  1039126  ? 


Find 

18.  lJ£/Mof$863.25. 

19.  1  sixth  of  34807  tons. 

20.  1  eighth  of  20673  days. 

21.  1  ninth  of  $7384.50. 


What  is 

22.  \  of  500322  miles  ? 

23.  i  of  32876  men  ? 

24.  £  of  60349  acres  ? 

25.  ^  of  760344  rods  ? 


26.  How  many  barrels  of  flour  at  $8  a  barrel,  can  be 
bought  for  $12736  ?    For  $7068  ? 

27.  If  75000  bushels  of  grain  are  put  into  8  bins  of 
equal  size,  how  many  bushels  does  each  bin  contain  ? 

28.  If  9  acres  of  land  cost  $976.50,  what  is  the  cost  of 
1  acre  ? 

29.  How  many  oranges  can  be  bought  for  $3.72,  at 
4  cents  a  piece  ? 

30.  At  8  cents  a  yard,  how  many  yards  of  ribbon  can 
be  bought  for  $7.28? 

31.  Paid  $1792  for  7  horses.    What  did  each  cost  ? 


60  DIVISION. 

ORATj     EXERCISES. 

134.  1.  The  quotient  of  two  numbers  is  15,  and  the 
divisor  is  8.     What  is  the  dividend  ? 

2.  The  dividend  is  96,  and  the  quotient  is  6.     What  is 
the  divisor  ? 

3.  The  quotient  is  12,  the  remainder  is   9,  and  the 
divisor  is  11.     What  is  the  dividend  ? 

4.  If  12  yards  of  cloth  cost  $35,  for  how  much  a  yard 
must  it  be  sold  to  gain  $13  ? 

5.  A  man  received  $50  for  5  barrels  of  pears,  and  paid 
all  but  $14  for  4  chairs.     What  did  each  chair  cost  ? 

6.  If  4  weeks'  board  cost  $28,  what  will  9  weeks'  board 
cost? 

Analysts. — One  week's  board  will  cost  1  fourth  of  $28,  or  $7 ; 
and  9  weeks'  board  will  cost  9  times  $7,  or  $63. 

7.  If  8  yards  of  silk  cost  $32,  what  will  12  yards  cost  ? 

8.  What  will  15  sheep  cost,  if  5  sheep  cost  $35  ? 

9.  How  many  cords  of  wood  at  $4  dollars  a  cord,  will 
pay  for  6  barrels  of  flour  at  $8  a  barrel  ? 

Analysis. — Six  barrels  of  flour  will  cost  6  times  $8,  or  $48  ;  and  $4, 
the  price  of  1  cord  of  wood,  are  contained  in  $48, 12  times.    Hence,  etc. 

10.  How  many  days'  labor  at  $4  a  day  will  pay  for  3  tons 
of  coal  at  $6  a  ton,  and  2  tons  of  hay  at  $15  a  ton  ? 

11.  How  many  pounds  of  meat  at  12  cents  a  pound,  will 
( ost  as  much  as  9  pounds  of  cheese  at  8  cents  a  pound  ? 

Complete  the  following  equations  : 

12.8x0  +  6x4^8=? 
13.10x12—0x6-^6=? 


14.9x11-54-7-6  +  20=? 


15.  63-f-7x0  +  12=? 

16.  (108-r-12)xll—  25=? 

17.  90-18-j-(44-7x6)  =  : 


DIVISION.  61 

WRITTEN    EXERCISES. 

135.  When  the  divisor  consists  of  more  than  one 
order  of  units. 

1.  Divide  5437  by  26. 

operation.  Analysis. — 26  is  contained    in    54 

Divisor.  Dividend.  Quotient.        hundreds,  2  hundred  times,  with  a  re- 

26)5437(209^-      mainder.    Write  the  2  hundreds    in 

5  2  the  quotient,  and  multiply  the  divisor 

26  by  this  quotient  figure,  and  subtract 

the  product,    52    hundreds,    from  54 

2  3  4  hundreds,  the  first  partial  dividend, 

3  R  mainder        an^  there   rtmains   2   hundreds.     To 

this  annex  the  3  tens  of  the  dividend, 

making  23  tens  for  the  second  partial  dividend. 

26  is  not  contained  in  23,  so  write  a  cipher  in  the  quotient  and 
bring  down  the  7  units  of  the  dividend,  making  237  units  for  the 
third  partial  dividend. 

26  is  contained  in  237  units  9  times,  with  a  remainder.  Write 
the  9  units  in  the  quotient,  and  multiplying  and  subtracting  as 
before,  there  remain  3  units,  which  write  over  the  divisor,  and 
annex  as  a  part  of  the  quotient.    Hence  the  quotient  is  209^. 

136.  Long  Division  is  the  process  of  dividing 
when  the  subtractions  are  written. 

2.  Find  how  many  times  204  is  contained  in  1041835. 

OPERATION.  *  PROOF. 

Divisor.        Dividend.       Quotient. 
204)1041835(5107^  5107   Quotient. 

10  2  0  2  0  4  Divisor. 


218  20428 

204  10214 


1435 

1041828 

1428 

7   Remainder. 

7  Remainder, 

10418  3  5   Dividend. 

G3  division.. 

3.  Divide  32762  by  14  ;    by  16  ;    by  23  ;    by  28. 

4.  Divide  130426  by  58  ;    by  63  ;    by  81 ;    by  74. 

KiTLE. — I.  Write  the  divisor  at  the  left  of  the  dividend, 
with  a  line  between  them. 

II.  Find  how  many  times  the  divisor  is  contained  in  the 
least  number <of  the  left  hand  orders  of  the  dividend  that 
will  contain  it,  and  write  the  result  for  the  first  figure 
of  the  quotient. 

III.  Multiply  the  divisor  by  this  quotient  figure,  sub- 
tract the  product  from  the  partial  dividend  used,  and  to 
the  remainder  annex  the  figure  of  the  next  lower  order  of  the 
dividend  for  a  new  partial  dividend,  and  divide  as  before. 

IV.  Proceed  in  the  same  manner  until  all  the  orders  of 
the  dividend  have  been  used. 

V.  If  any  partial  dividend  does  not  contain  the  divisor, 
write  a  cipher  in  the  quotient,  and  annex  the  next  order  of 
the  dividend,  and  proceed  as  before. 

VI.  If  there  be  at  last  a  remainder,  write  it  after  the 
quotient  with  the  divisor  underneath. 

Proof. — Multiply  the  divisor  by  the  quotient,  and  to 
the  product  add  the  remainder,  if  any.  If  the  ivork  is 
correct,  the  result  ivill  be  equal  to  the  dividend. 

1.  If  the  product  of  the  divisor  and  quotient  be  greater  than  the 
partial  dividend,  the  quotient  is  too  large,  and  must  be  diminislted. 

2.  If  any  remainder  is  equal  to  or  greater  than  the  divisor,  the 
quotient  is  too  small  and  must  be  increased. 

137.  When  the  divisor  and  dividend  are  both  concrete 
numbers,  they  must  be  of  the  same  name.  Hence,  if  one 
be  dollars,  and  the  other  cents,  or  dollars  and  cents,  before 
dividing,  change  so  that  both  may  be  cents. 


DIVISION 


63 


138.  Since  100  cents  make  1  dollar,  there  are  100  times 
as  many  c^nts  as  dollars.     Hence, 

To  change  a  number  representing  dollars  to  a  number  represent- 
ing cents,  annex  two  ciphers  (110),  omit  the  sign  ($)  and  write  the 
word  cents  after  it. 

To  change  dollars  and  cents  to  the  same  form,  omit  the  sign  ($) 
nd  the  point  ( . )  and  write  the  word  cents  at  the  right. 


5.  Divide  $46.92  by  23. 

6.  Divide  $46.92  by  23  cents. 

OPERATION. 

OPERATION. 

23)$46.92($2.04 

23)4692(204times. 

46 

46 

92 

92 

92 

92 

7.  Divide  $46.92  by  $23.        8.  Divide  $46  by  23  cents. 

OPERATION.  OPERATION. 

2300(4692(2^^  times.  *  23)4600(200  times. 
4600  46 


92 


00 


In  like  manner  divide,  and  prove  the  following  : 


9. 

$325.72  by  34. 

14. 

$288.96  by  $.43. 

10. 

$938.07  by  63. 

15. 

$810.98  by  $.46. 

11. 

$3176.46  by  126. 

16. 

$594  by  18  cents 

12. 

$49.56  by  14  cents. 

17. 

$1385  by  $105. 

13. 

$87.36  by  21  cents. 

18. 

$32.48  by  $7. 

How  many  times 

19.  Is  47  contained  in  30176  ?   In  27865  ?   In  103474  ? 

20.  Is  185  contained  in  200376  ?    In  4701625  ? 

21.  The  annual  receipts  of  a  company  are  $570685. 
What  is  the  average  a  day,  if  there  are  313  working  days  ? 


64 


DIVISION 


22.  If  .867  shares  of  railroad  stock  are  valued  at  $84099, 
what  is  the  value  of  each  share  ? 

23.  A  plantation  of  736  acres  was  sold  for  $55936. 
What  was  the  price  of  an  acre  ? 

24.  Paid  $17100  for  a  farm,  at  the  rate  of  $36  an  acre. 
How  many  acres  did  it  contain  ? 

25.  How  many  horses,  at  $125  each,  will  $4735  buy, 
and  how  much  money  will  be  left  ? 


Divide 

Divide 

26. 

33490  by  85. 

34. 

863256  by  736. 

27. 

740070  by  135. 

35. 

1646301  by  381. 

2.8. 

1554768  by  216. 

36. 

5226412  by  2567. 

29. 

5497800  by  175. 

37. 

11214887  by  3076. 

30. 

3931476  by  556. 

38. 

75862500  by  10115. 

31. 

5120401  by  $87. 

39. 

313194105  by  7153. 

32. 

1018090  by  1669. 

40. 

1246038849  by  269181. 

33. 

73484248  by  2621 

41. 

2331883954  by  6739549 

139.  To  divide  by  the  factors  of  a  number. 


1.  Divide  644  by  28,  using  the  factors. 


OPERATION. 

4)644 


7)161 


23 


Analysis. — Since  28  is  equal  to  4  times  7,  divide 
either  by  28,  or  by  its  factors  4  and  7.  Now,  644 -f- 
4=161  ;  but  this  quotient  is  7  times  too  great,  and 
must  therefore  be  divided  by  7  ;  hence,  161-^7=23, 
the  true  quotient. 


Factors. 

2.  Divide      6228  by    36,  or  by  4,  and  9. 

3.  Divide    27360  by    96,  or  by  3,  4,  and  8. 

4.  Divide  526050  by  126,  or  by  2,  7,  and  9. 

5.  Divide    73416  by  168,  or  by  4,  6,  and  7. 


DIVISION.  .65 

6.  Divide  5831  by  84,  using  the  factors,  3,  4,  and  7. 

operation.  Analysis.  —  Since    84   is 

3)5831  •                                        equal  to  3x4x7,  divide  by 

84,  or  by   its  factors  3,  4, 

4)1943 2       and  7. 

7)485  .      .      .3x3=     9          5831  -=-  3  =  1943,  and  a  re- 

TT  n       4      o        r>  a       mainder  of  2,  which  being  a 

69     .     .2x4x3  =  24  ..».,     ,.  .,     ,.     , 

part  of  the  dividend,  is  also 

True  Remainder.    3  5       a  part  of  the  true  remainder. 

0  a  , ,  1943  -i-  4  =  485,  and  a  re- 

6  9  |i   Quotient.  .    ,        M  _         ' 

84  mamder  ot  3.     Since  a  unit 

of  the  first  quotient  1943,  equals  3  units  of  the  dividend,  this  second 
remainder  3  being  a  part  of  1943,  equals  3x3,  or  9  units  of  the 
dividend. 

485  ■*■  7  =  69,  and  a  remainder  of  2.  Since  a  unit  of  the  second 
quotient  485  equals  4  units  of  the  first  quotient  1943,  this  third  re- 
mainder 2  being  a  part  of  485,  equals  2  x  4  x  3,  or  24  units  of  the 
dividend.  Hence  the  first  partial  remainder  is  2,  the  second  is  9, 
the  third  is  24,  and  the  true  remainder  35  ;  and  the  quotient  69|f. 

7.  Divide  139074  by  72,  using  its  factors  3,  4,  and  6. 

8.  Divide  7360479  by  96,  using  its  factors  2,  6,  and  8. 

Rule. — I.  Separate  the  divisor  into  two  or  more  factors. 

II.  Divide  the  dividend  by  one  of  these  factors,  and  the 
quotient  thus  obtained  by  another  factor,  and  so  on  until 
all  the  factors  have  been  used  as  divisors. 

III.  If  there  be  remainders,  multiply  each  remainder  by 
all  the  divisors  preceding  the  one  that  produced  it. 

IV.  Add  the  products  and  the  remainder  from  the  first 
division,  if  any,  and  the  sum  will  be  the  true  remainder, 

9.  Divide  376875  by  315,  or  by  5,  7,  and  9. 

10.  Divide    73522  by  135,  or  by  3,  5,  and  9. 

11.  Divide  401976  by  245,  or  by  5,  7,  and  7. 


66 


DIVISION. 


140.  When  the  divisor  has  ciphers  on  the  right. 

1.  Divide  4067  by  10. 


OPERATION. 

110)40617 
406  . 


Analysis. — Since  remoVing  any  order 
of  figures  one  place  to  the  right,  dimin- 
ishes its  value  ten  times  (45),  by  cutting 
off,  or  taking  away,  the  right-hand  fig- 
ure of  a  number,  each  of  the  remaining 
figures,  being  removed  one  place  to  the 

right,  is  diminished  in  value  ten  times,  or  divided  by  10. 

For  similar  reasons,  cutting  off  two  figures  divides  by  100,  cutting 

off  three  figures,  divides  by  1000,  and  so  on.     The  remaining  figures 

are  the  quotient,  and  those  cut  off,  the  remainder. 


7  Rem. 
4  0  6  y'k  Quotient. 


Divide 

2.  37684  by  100. 

3.  103076  by  1000. 

6.  Divide  2416700  by  6000 


4.  267104  by  10000. 

5.  5023001  by  100000. 


OPERATION. 

61000)24161700 


Analysis — Resolve  6000 
into  the  factors  1000,  and  6. 
First  divide  by  1000,  by  cut- 
ting ofZ  the  three  rignt-hand 
figures  of  the  dividend.  The 
quotient  is  2416,  and  a  re- 
mainder of  700.  Next  divide  2416  by  6  ;  the  quotient  is  402  and  a 
second  remainder  of  4  thousands,  which  prefixed  to  the  first  re- 
mainder 700  gives  a  true  remainder  of  4700.    Hence  the  quotient 


402     .     .     4700  Rem. 
40  2  Wot  Quotient. 


In  like  manner,  divide 

7.  307200  by      900.  9.     5761321  by    2040. 

8.  7820305  by  28000.  10.     8073160  by  14800. 

Kule. — I.   Cut  off  the  ciphers  from  the  right  of  the 
divisor,  and  as  many  figures  from  the  right  of  the  dividend, 

II.  Divide  the  remaining  part  of  the  dividend  by  the 
remaining  part  of  the  divisor. 


division.  67 

III.  Prefix  the  remainder,  if  any,  to  the  figures  cut  off, 
and  the  result  will  be  the  true  remainder. 

11.  If  it  require  $34400  to  pay  a  regiment  of  800  men, 
how  much  does  each  man  receive  ? 

12.  At  $3400,  how  many  lots  can  be  bought  for  $68000  ? 

13.  How  many  bales,  each  weighing  470  pounds,  can 
be  made  of  39500  pounds  of  cotton  ? 

GENERAL    PRINCIPLES    OF    DIVISION. 

141.  The  quotient  depends  upon  the  relative  values  of 
the  dividend  and  divisor.  Hence,  any  change  in  the 
value  of  either  dividend  or  divisor,  will  produce  a  change 
in  the  value  of  the  quotient.  But  some  changes  may  be 
made  upon  both  dividend  and  divisor,  which  will  not 
affect  their  relative  values,  and  consequently  will  not 
affect  the  quotient.  To  illustrate,  let  54-f-9=6,  be  the 
fundamental  equation,  with  which  the  following  are  to 
be  compared  : 

1.  (54x3)^9  =  162-^9  =  18.   Multiplying  the  dividend  by  3 

multiplies  the  quotient  by  3. 

2.  54_i_(9_i_3)  —54-^3 =18.      Dividing  the  divisor  by  3  mul- 

tiplies the  quotient  by  3. 

3.  (54_^3)_i_9  — 18-^9  =  2.        Dividing  the  dividend  by  3  di- 

vides the  quotient  by  3. 
I.  54-r-  (9  X  3)  ==54-r-27  =  2.     Multiplying  the  divisor  by  3  di- 
vides the  quotient  by  3. 
6.  (54  X  3)  -T-  (9x3)  =  162-f-      Multiplying  both  dividend  and 
27  =  6.  divisor  by  3  does  not  change 

the  quotient. 
£   (54_^_3)_i.(9-t-3)  =  18-v-        Dividing  both  dividend  and  di- 
3_- g#  visor  by  3  does  not  change  the 

quotient. 


68  division. 

These  six  equations  illustrate  the  following 

142.  General  Principles  of  Division. 

1.  Multiplying  the  dividend,  or       )  Multiplies  the  quo* 
Dividing  the  divisor,  )      tient. 

2.  Dividing  the  dividend,  or  )  Divides    the  quo- 
Multiplying  the  divisor,  )      tient. 

3.  Multiplying   or    dividinq   loth  \  ^  ,     , 

-,-■-,     n       -,    7.  •        7     ,7    /  Does    not   change 
dividend  and  divisor  by  the  V      ,T  ,.     , 

(      the  guotient. 
same  number,  )  * 

These  three  principles  may  be  embraced  in  one 

GENERAL   LAW. 

143.  A  change  in  the  dividend  produces  a  like  change 
in  tlie  quotient,  but  a  change  in  the  divisor  produces  an 
opposite  change  in  the  quotient. 

GENERAL    REVIEW. 
OBAZ    EX  AMP  LBS. 

144.  1.  The  sum  of  three  numbers  is  40.  One  of  the 
numbers  is  12,  and  another  is  15.     What  is  the  third  ? 

2.  The  difference  of  two  numbers  is  16,  and  the  smaller 
is  12.    What  is  the  larger  ? 

3.  The  difference  of  two  numbers  is  18,  and  the  larger 
is  30.     What  is  the  smaller  ? 

4.  The  product  of  two  numbers  is  132,  and  one  of  the 
numbers  is  11.     What  is  the  other  ? 

5.  What  five  numbers  less  than  10  will  divide  120 
without  a  remainder  ? 

6.  The  sum  of  two  numbers  is  21,  and  the  greater  12. 
What  is  the  product  of  the  two  numbers  ? 


REVIEW.  69 

7.  The  quotient  of  two  numbers  is  45,  and  the  divisor 
8.    What  is  the  dividend  ? 

8.  How  many  times  can  8  bushels  of  grain  be  taken 
from  a  bin  containing  52  bushels,  and  what  will  remain  ? 

9.  A  news-boy  sold  24  papers  at  4  cents  each,  and  there- 
by gained  48  cents.    At  what  rate  did  he  buy  the  papers  ? 

10.  The  dividend  is  240  and  the  quotient  12.     What  is 
the  divisor  ? 

11.  The  quotient  is  20,  the  remainder  8,  and  the  divi- 
sor 9.     What  is  the  dividend  ? 

12.  A  drover  bought  10  sheep  at  $8  a  head,  and  sold 
them  for  $96.     How  much  did  he  gain  a  head  ? 

How  many 

13.  In  each  of  5  equal  parts  of  (9  x  12—8  x  6)  ? 

14.  In  each  of  9  equal  parts  of  (56—0  x  7  +  16)  ? 

15.  In  each  of  7  equal  parts  of  (72—40  +  37—20)  ? 

16.  If  5  men  can  build  a  wall  in  9  days,  in  how  many 

days  can  3  men  build  it  ? 

Analysis. — It  will  take  1  man  5  times  9  days,  or  45  days  ;  and 
3  men  can  build  it  in  I  third  of  45  days,  or  15  days. 

17.  How  long  will  it  take  7  men  to  do  the  same  work 
that  14  men  can  perform  in  3  days  ? 

18.  If  9  days'  work  will  pay  for  6  tons  of  coal  at  $6  a 
ton,  what  is  the  price  of  a  day's  labor  ? 

19.  How  much  pork  can  be  bought  for  96  cents,  if  9 
pounds  cost  72  cents  ? 

20.  If  5  men  can  build  a  wall  in  8  days,  how  many 

men  can  build  it  in  4  days  ? 

Analysis. — It  will  require  8  times  5  men,  or  40  men,  to  build 
it  in  1  day,  and  1  fourth  of  40  men,  or  10  men,  to  build  it  in  4  days. 


70  DIVISION. 

21.  How  many  men  will  be  required  to  do  the  same 
work  in  5  days  that  4  men  can  do  in  40  days  ? 

22.  If  6  men  can  dig  a  ditch  in  5  days,  how  many  men 
would  be  required  to  dig  it  in  1  day  ?  In  2  days  ?  In  3 
days  ?     In  6  days  ?     In  10  days  ? 

23.  At  the  rate  of  24  miles  in  8  hours,  how  many  miles 
would  a  man  walk  in  12  hours  ? 

24.  If»  a  woman  pay  60  cents  for  some  lemons,  at  the 
rate  of  10  cents  for  6,  and  sell  them  at  the  rate  of  9  for 
20  cents,  how  many  cents  will  she  gain  ? 

25.  If  5  barrels  of  flour  are  worth  $60,  how  many 
cords  of  wood  at  $4  a  cord  will  pay  for  3  barrels  ? 

26.  If  12  yards  of  cloth  cost  $40,  for  how  much  must 
it  be  sold  a  yard  to  gain  $20  ? 

27.  What  cost  9  quarts  of  milk,  if  4  quarts  cost  24  cents? 

28.  How  many  bags  will  be  required  to  hold  108  bushels 
of  wheat,  if  4  bags  hold  9  bushels  ? 

29.  To  6  add  8,  subtract  4,  multiply  by  5,  add  6, 
divide  by  8,  and  what  is  the  result  ? 

30.  How  much  greater  is  7  times  8  plus  4,  than  72 
divided  by  9,  multiplied  by  7  ? 

31.  How  much  less  is  10  times  10,  diminished  by  4 
times  10,  plus  12,  than  100  divided  by  10,  plus  8  times  11  ? 

Find  the  required  term  in  the  following  epuations  : 
32.  25  +  9—32  +  4=? 


33.  4x12  +  3x9=? 

34.  60—124-6  x  ?  =56 

35.  72-^-9  x22=10=? 

36.  1204-20  +  484-?  =9 

37.  96^8x9=?  x!2 


38.  (32  +  12h-H)x  ?  =80 

39.  (1324-11— 4)  x  9  =  60  +  ? 

40.  42  +  24—15=?  +10 

41.  48  +  364-48—36  =  16-? 

42.  (120— 7x12)^-6=?  4-11 

43.  49  +  144-(28— 19)=25- ? 


REVIEW.  71 

WRITTEN     EXAMPLES. 

145.  1.  Subtract  2520  from  the  sum  of  3472,  450, 1254, 
and  56  ;  divide  the  remainder  by  113,  and  multiply  the 
quotient  by  205.     What  is  the  result  ? 

2.  How  many  times  can  236  be  subtracted  from  2124  ? 

3.  How  many  times  236  will  produce  2124  ? 

4.  The  factors  of  a  number  are  36  -f  114,  and  5640 
—  3007.     What  is  the  number  ? 

5.  The  product  of  two  numbers  is  30128,  and  one  of 
the  numbers  is  4200  -^  75.     What  is  the  other? 

6.  Divide  the  product  of  204  and  378  by  their  difference. 

7.  What  must  be  added  to  the  sum  of  $12.36  and  $7.62, 
to  amount  to  $30.76? 

8.  What  is  the  difference  between  746  x  23  and  18975 
^25? 

9.  A  man  owing  a  debt  of  $3000,  paid  $756.50  at  one 
time,  $1289.75  at  another,  and  then  made  a  third  pay- 
ment large  enough  to  reduce  the  debt  to  $925.60.  What 
was  the  third  payment  ? 

10.  How  many  pounds  of  butter  at  40  cents  a  pound  are 
worth  as  much  as  1600  bushels  of  oats  at  75  cents  a  bushel  ? 

11.  If  a  man  gain  $638.75  by  selling  365  barrels  of 
flour  at  $9.25  a  barrel,  at  what  price  did  he  buy  it  ? 

12.  The  multiplier  is  36,  and  the  product  170352 ;  if 
the  multiplier  is  1  fourth  as  great,  what  is  the  product? 

13.  The  multiplier  is  204,  and  the  multiplicand  is 
17605  ;  if  the  multiplicand  were  one-fifth  as  great,  what 
would  be  the  product  ?  ( 

14.  If  a  mechanic  receives  $1500  a  year  for  his  labor, 
and  his  expenses  are  $968,  in  what  time  can  he  save 
enough  to  buy  28  acres  of  land  at  $133  an  acre  ? 


72  DIVISION. 

15.  With  the  multiplier  48,  the  product  is  166656;  with 
a  multiplicand  1  third  as  great,  what  would  be  the  product  ? 

16.  The  divisor  is  16,  the  quotient  12624;  with  a  divi- 
sor 1  fourth  as  great,  what  would  be  the  quotient  ? 

17.  The  divisor  is  24,  and  the  quotient  is  43950  ;  if  the 
divisor  be  made  6  times  as  large,  what  will  be  the  quotient  ? 

18.  The  quotient  is  91864 ;  with  a  divisor  1  ninth  as 
great,  what  would  be  the  quotient  ? 

19.  A  grocer  bought  two  kinds  of  syrup;  one  for  54 
cents  a  gallon,  and  the  other  for  62  cents.  What  was  the 
average  cost  a  gallon  ? 

Operation.— (54  cents  +  62  cents)  -h  2  =  58  cents. 

The  average  of  two  numbers  is  one-half  their  sum,  the  average  of 
three  numbers  is  one-third  their  sum,  etc. 

20.  A  merchant  bought  equal  quantities  of  3  kinds  of 
tea,  some  at  60  cents,  some  at  78  cents,  and  some  at 
90  cents  a  pound.     What  was  the  average  cost  a  pound  ? 

21.  A  keeper  of  a  toll  bridge  received  $104  toll  on 
Monday,  $97  on  Tuesday,  $128  on  Wednesday,  and  $99 
on  Thursday.     What  were  the  average  daily  receipts  ? 

22.  Sold  3  city  lots  for  $1500,  $2976,  and  $1895,  respec- 
tively.    What  was  the  average  price? 

23.  If  a  young  man  receive  a  salary  of  $25  a  week,  and 
he  pays  $8.75  for  his  board,  and  $4.65  for  other  expenses, 
in  how  many  weeks  can  he  pay  a  debt  of  $487.20  ? 

24.  A  man  having  $4578  paid  out  all  but  $1642  in  S 
weeks.    What  was  the  average  amount  paid  out  each  week  ? 

25.  Bought  140  acres  of  land  for  $7560,  and  sold  86 
acres  of  it  at  $75  an  acre,  and  the  remainder  at  cost. 
How  much  was  gained  ? 


REVIEW.  73 

26.  A  father  gave  his  property  to  his  4  children.  To 
the  first  he  gave  $6780,  to  the  second  $8200,  to  the  third 
$1526  more  than  to  the  first,  and  to  the  fourth  $1345  less 
than  to  the  third.     What  was  the  value  of  his  property  ? 

27.  The  sum  of  two  numbers  is  184,  and  their  differ- 
ence is  42.     What  are  the  numbers  ? 

Analysis. — Since  184  is  the  sum  of  the  numbers,  if  the  differ- 
ence 42  be  subtracted  from  the  sum  184,  the  remainder  142  will  be 
twice  the  less  number.  142  -f-  2  =  71  the  less  number  ;  and  71  -l-  42 
=  113  the  greater  number. 

Or,  if  the  difference  42  be  added  to  the  sum  184,  the  amount  226, 
will  be  twice  the  greater  number.  226  -f2=  113  the  greater  num- 
ber ;  and  113  —  42  =  71  the  less  number. 

Proof.— 113  +  71  =  184  the  sum. 

28.  The  sum  of  two  numbers  is  5672,  and  their  differ- 
ence is  1974.     What  are  the  numbers  ? 

29.  A  man  paid  $1250  for  a  horse  and  carriage,  the 
horse  being  valued  at  $190  more  than  the  carriage.  What 
was  the  value  of  each  ? 

30.  At  a  town  election  the  whole  number  of  votes  cast 
for  two  candidates  was  3789,  and  the  majority  for  the 
successful  candidate  was  227.  How  many  votes  did  each 
receive  ? 

31.  Two  men  are  worth  $28475,  and  one  is  worth  $4625 
more  than  the  other.     How  much  is  each  man  worth  ? 

32.  A  grocer  wishes  to  put  240  pounds  of  tea  into  three 
kinds  of  boxes,  containing  respectively  5,  10,  and  15 
pounds,  using  the  same  number  of  boxes  of  each  kind. 
How  many  boxes  will  be  required  ? 

33.  Sold  a  quantity  of  wood  for  $2492,  that  cosi  $1424, 
thus  gaining  $3  a  cord.  How  many  cords  were  there, 
and  what  was  the  cost  per  cord  ? 


74  DIVISION. 

34.  What  number  divided  by  36,  the  quotient  increased 
by  48,  the  sum  diminished  by  37,  the  remainder  multiplied 
by  14,  and  the  product  increased  by  216  -i-  72,  is  269  ? 

Find  the  missing  term  in  the  following  equations  : 

35.  (15341-^-29)  x  (8430-^-1405)  =  1587  x  ? 

36.  [4500  +  (12000  -1375)-=- 121  x  25]  x  48=  ?  x24 

37.  732  x  6-^(15  X  24-^9xJL0)  +  (42  x  234-^26)  =  ? 

38.  450  +  (24-12)x 5-^(90^6) -f  (3^<lT-18)=? 

146.  The  pupil  should  illustrate  the  following  prob- 
lems by  original  examples : 

Problem  1.  Given  several  numbers,  to  find  their  sum. 

2.  Given  the  sum  of  several  numbers  and  all  of  them 
but  one,  to  find  that  one. 

3.  Given  the  parts,  to  find  the  whole. 

4.  Given  the  whole  and  all  the  parts  but  one,  to  find 
that  one. 

5.  Given  two  numbers,  to  find  their  difference. 

6.  Given  the  greater  of  two  numbers  and  their  differ- 
ence, to  find  the  less. 

7.  Given  the  less  of  two  numbers  and  their  difference, 
to  find  the  greater. 

8.  Given  the  minuend  and  subtrahend,  to  find  the  re- 
mainder. 

9.  Given  the  minuend  and  remainder,  to  find  the  sub- 
trahend. 

10.  Given  the  subtrahend  and  remainder,  to  find  the 
minuend. 

11.  Given  two  or  more  numbers,  to  find  their  product. 


REVIEW.  75 

12.  Given  the  product  and  one  of  two  factors,  to  find 
the  other  factor. 

?.3.  Given  the  multiplicand  and  multiplier,  to  find  the 
product. 

14.  Given  the  product  and  multiplicand,  to  find  the 
multiplier. 

15.  Given  the  product  and  multiplier,  to  find  the  mul- 
tiplicand. 

16.  Given  two  numbers,  to  find  their  quotient. 

17.  Given  the  divisor  and  dividend,  to  find  the  quotient. 

18.  Given  the  divisor  and  quotient,  to  find  the  dividend. 

19.  Given  the  dividend  and  quotient,  to  find  the  divisor. 

20.  Given  the  divisor,  quotient,  and  remainder,  to  find 
the  dividend. 

21.  Given  the  dividend,  quotient,  and  remainder,  to 
find  the  divisor. 

22.  Given  the  final  quotient  of  a  continued  division 
and  the  several  divisors,  to  find  the  dividend. 

23.  Given  the  quotient  of  a  continued  division,  the  first 
dividend,  and  all  the  divisors  but  one,  to  find  that  divisor. 

24.  Given  the  dividend  and  several  divisors  of  a  con- 
tinued division,  to  find  the  quotient. 

25.  Given  two  or  more  sets  of  numbers,  to  find  the 
difference  of  their  sums. 

26.  Given  two  or  more  sets  of  factors,  to  find  the  sum 
of  their  products. 

27.  Given  two  or  more  sets  of  factors,  to  find  the  dif- 
ference of  their  products. 

28.  Given  the  sum  and  the  difference  of  two  numbers, 
to  find  the  numbers. 


76  division. 

147.  SYNOPSIS  FOE  REVIEW. 

f  1.  Division.     2.  Dividend.     3.  Divi. 

1.  Definitions.  <       sor.    4.  Quotient.    5.  Remainder, 
|^       6.  Sign  of  Division. 

2.  Principles,  1,  2,  and  3. 

3.  Relation  of  Division  to  Subtraction. 

4.  Relation  of  Division  to  Multiplication. 

Illustrate. 


5.  Objects  of  Division 
G.  Equal  Parts. 
7.  Short  Division 


■r, 

l, 


Definition. 
Method. 


8.  Long  Division. 


1.  Definition. 

2.  Method. 

3.  Rule,  I— VI. 

4.  Proof. 


r  1.  When  divisor  and  dividend 
are  concrete,  hut  unlike. 

2.  How   to    change    dollars    to 
cents. 

3.  How  to  change  dollars  and 
cents  to  cents. 


9.  Division  of  Dollars 


1 


Method. 
2.  Rule,  I,  II,  III,  IV. 


10.  Division  by  Factors.  < 

11.  When  the  Divisor  has  Ciphers  J  1.  Method. 

on  the  Right.  (  2.  Rule,  I,  II,  ILL 

12.  General  Principles  of  Division,  1,  2,  3. 

13.  General  Law. 


PllFlKIiSfglBMlEBI 

148.  1.  What  two  numbers,  besides  the  number  itself 
and  1,  will  give  a  product  of  8  ?     16  ?     25  ?    42  ?     64  ? 

2.  What  numbers,  other  than  the  given  number  and  1, 
will  exactly  divide  9  ?     15?    36?    48?     55? 

3.  Of  what  sets  of  two  numbers  is  24  the  product  ? 

4.  Of  what  sets  of  three  numbers  is  36  the  product  ? 

5.  What  are  the  smallest  numbers,  other  than  1,  that 
will  exactly  divide  18?    21?    49?    55? 

6.  What  is  the  largest  number,  other  than  the  given 
number  itself,  that  will  exactly  divide  22  ?  24  ?  30  ?  40  ? 

7.  Name  the  numbers  between  12  and  30,  that  are  the 
product  of  two  factors  greater  than  1.     Between  30  and  50. 

8.  Name  the  numbers  between  5  and  20,  that  have  no 
other  factors  than  the  numbers  themselves  and  1. 

9.  Of  what  number  are  7  and  8  the  factors  ?    2,  5,  and 
7?    4,  5,  and  3?    2,  3,  5,  and  10? 

DEFINITIONS. 

149.  The  Properties    of  Numbers  are  those 
qualities  or  elements  which  necessarily  belong  to  numbers. 

Numbers  are  either  Integral,  Fractional,  or  Mixed. 

150.  An  Integral  Number  or  Integer  is  a 

number  representing  whole  things.     (4.) 

Thus,  8,  23,  30  men,  45  pounds  are  integral  numbers. 
Integral  numbers  are  either  Even  or  Odd,  Prime  or  Composite. 


78  PROPERTIES     OF     NUMBERS. 

151.  An  Even  Number  is  a  number  that  is  exactly 
divisible  by  2. 

All  numbers  whose  unit  figure  is  0,  2,  4,  6,  or  8,  are  even. 

152.  An  Odd  Number  is  a  number  that  is  not 
exactly  divisible  by  2. 

All  numbers  whose  unit  figure  is  1,  3,  5,  7,  or  9,  are  odd. 

153.  A  Prime  Number  is  a  number  that  has  no 
integral  factors  except  unity  and  itself. 

Thus,  2,  3,  5,  11,  23,  etc.,  are  prime  numbers. 
2  is  the  only  even  prime  number. 

154.  A  Composite  Number  is  a  number  that  has 
other  integral  factors  besides  unity  and  itself. 

Thus,  21  is  a  composite  number,  since  21  =  7  x  3. 

155.  The  Factors  of  a  number,  are  the  numbers 
which  multiplied  together  will  produce  it.     (108.) 

Thus,  7  and  8  are  factors  of  56  ;  3,  4,  and  7,  of  84. 

156.  A  Prime  Factor  is  a  prime  number  used  as 
a  factor.    (153.) 

The  prime  factors  of  a  number  are  also  the  prime  divisors  oi  it. 

157.  An  Exact  Divisor  of  a  number  is  one  that 
will  divide  that  number  without  a  remainder. 

Thus,  6  is  an  exact  divisor  of  48,  and  9  an  exact  divisor  of  72. 

1.  The  Exact  Divisors  of  a  number  are  also  the  factors  of  that 
number. 

2.  An  exact  divisor  of  a  number  is  sometimes  called  the  measure 
of  that  number. 

3.  When  a  number  is  a  factor,  or  divisor,  of  each  of  two  or  more 
numbers,  it  is  called  a  common  factor,  or  divisor,  of  those  numbers. 

158.  Numbers  are  prime  to  each  other  when  they 
have  no  common  integral  factors,  or  divisors. 

Thus,  9  and  14,  16  and  25  are  prime  to  each  other. 


DIVISIBILITY     OF     NUMBERS.  79 

DIVISIBILITY    OF    NUMBERS. 

159.  A  number  is  said  to  be  divisible  by  another, 
when  there  is  no  remainder  after  dividing.  Any  number 
is  divisible 

1.  By  2,  if  it  is  an  even  number. 
Thus,  20,  24,  86,  and  44  are  divisible  by  2. 

2.  By  3,  if  the  sum  of  its  digits  is  divisible  by  3. 
Thus,  135,  471,  and  1134  are  divisible  by  3. 

3.  By  4,  if  its  two  right-hand  figures  are  ciphers,  or 
express  a  number  divisible  by  4. 

Thus,  300,  432,  and  1548  are  divisible  by  4. 

4.  By  5,  if  it  ends  with  a  cipher  or  5. 
Thus,  30,  45,  and  235  are  divisible  by  5. 

5.  By  0,  if  it  is  an  even  number  and  divisible  by  3. 
Thus,  168,  402,  and  1314  are  divisible  by  6. 

G.  By  8,  if  its  three  right-hand  figures  are  ciphers,  or 
express  a  number  divisible  by  8. 

Thus,  3000,  2728,  and  10576  are  divisible  by  8. 

7.  By  9,  if  the  sum  of  its  digits  is  divisible  by  9. 
Thus,  217683  and  401301  are  divisible  by  9. 

8.  By  10,  if  it  ends  with  one  or  more  ciphers. 

Thus,  40,  500,  3000  are  respectively  divisible  by  10,  100,  and  1000, 

9.  By  7,  11,  and  13,  if  it  consists  of  but  four  places, 
the  first  and  fourth  being  occupied  by  the  same  signifi- 
cant figures,  and  the  second  and  third  by  ciphers. 

Thus,  2002,  3003,  and  5005,  are  divisible  by  7,  11,  and  1& 


80  PROPERTIES     OF     NUMBERS. 

10.  An  odd  number  is  not  divisible  by  an  even  number. 

11.  If  an  even  number  is  divisible  by  an  odd  number, 
the  quotient  will  be  an  even  number. 

Thus,  the  quotient  of  36  divided  by  9,  is  4  ;  of  42  by  7,  is  6. 

12.  If  an  even  number  is  divisible  by  an  odd  number, 
it  is  also  divisible  by  twice  that  number. 

Thus,  28  is  divisible  by  7,  and  also  by  twice  7. 

13.  Every    odd   number   except   1,    increased  or  else 
diminished  by  1,  is  divisible  by  4. 

Thus,  11  increased  by  1,  or  17  diminished  by  1,  is  divisible  by  4 

14.  Every  prime  number  except  2   and  3,  increased 
or  else  diminished  by  1,  is  divisible  by  6. 

Thus,  23  increased  by  1,  or  31  diminished  by  1,  is  divisible  by  6. 

EXERCISES. 

160.  Find  by  inspection  some  of  the  exact  divisors  of 
the  following  numbers  : 

1.  1536.  4.     6105.  7.    32472. 

2.  1683.  5.     12936.  8.     71460. 

3.  3348.  6.     43560.  9.     197200. 

FACTORING. 

ORAE      EXERCISES. 

161.  1.  What  are  the  even  numbers  from  12  to  36  r 

2.  What  are  the  odd  numbers  from  12  to  36  ? 

3.  What  are  the  prime  numbers  from  12  to  36  ? 

4.  What  are  the  composite  numbers  from  12  to  36  ? 

5.  Name  all  the  prime  factors  of  36. 

6.  Name  all  the  composite  factors  of  36. 

7.  What  are  the  prime  factors  of  35  ?    49  ?    60  ? 

8.  What  are  the  composite  factors  of  32  ?    48  ?    72  ? 


FACTORING.  81 

9.  What  prime  factors  are  common  to  21  and  42  ? 

10.  What  composite  factors  are  common  to  36  and  72  ? 

11.  What  factors  are  common  to  18  and  30  ?    To  their 
sum  and  difference  f 

12.  What  factors  are  common  to  the  sum  and  difference 
of  20  and  40  ? 

13.  What  prime  factors  are  common  to  14  and  4  times  14? 

14.  What  two  composite  factors  are  common  to  24  and 
3  times  24  ? 

15.  What  is  the  largest,  and  what  the  smallest  prime 
factor  of  13,  30,  and  45  ? 

DEFINITIONS    AND    PEINCIPLES. 

162.  Factoring  is  the  resolving  of  a  composite 
number  into  its  factors,  and  is  performed  by  division. 

163.  An  Exponent  is  a  small  figure  written  at  the 
right  of  a  number,  and  a  little  above,  to  show  how  many 
times  the  number  is  used  as  a  factor. 

Thus,  23  —  2  x  2  x  2,  and  denotes  that  2  is  used  as  a  factor  3  times. 
54,  denotes  that  5  is  used  as  a  factor  4  times. 

164.  Principles  — 1.  The  prime  factors  of  a  number, 
or  the  product  of  any  two  or  more  of  them,  are  the  only 
exact  divisors  of  that  number. 

2.  A  factor  of  a  number  is  a  factor  also  of  any  number 
of  times  that  number. 

3.  A  factor  common  to  two  or  more  numbers  is  a  factor 
of  their  sum,  and  also  of  the  difference  of  any  tivo  of 
them. 

4.  Every  composite  number  is  equal  to  the  product  of  its 
prime  factors. 


82 


PKOPERTIES     OF     NUMBERS. 


WRITTEN     EXERCISES. 

165»  To  find  all  the  prime  factors  of  a  composite 
number. 

1.  What  are  the  prime  factors  of  2772  ? 


OPERATION. 

2)2772 

2)1386 

3)693 

3)231 

7)77 
11 


Analysis.— Since  the  given  number  is  even,  di- 
vide it  by  2,  the  least  prime  factor,  and  the  result 
also  by  2,  which  gives  an  odd  number  for  a  quotient. 

Next  divide  by  the  prime  factors  3,  3,  and  7,  suc- 
cessively, obtaining  for  the  last  quotient  11,  which 
not  being  divisible,  is  a  prime  factor  of  the  given 
number.  Hence  the  divisors  2,  2,  3,  3,  7,  and  the 
last  quotient  11,  are  all  the  prime  factors,  or  divisors, 
of  2772,  and  may  be  written  22,  32,  7,  11. 


In  like  manner  find  the  prime  factors  or  divisors 


2.  Of  1050. 

3.  Of  1140. 


4.  Of  2445. 

5.  Of  2366. 


6.  Of  2205. 

7.  Of  2310. 


Rule. — Divide  the  given  number  by  any  prime  factor 
of  it,  and  the  resulting  quotient  by  another,  and  so  continue 
the  division  until  the  quotient  is  a  prime  number.  The 
several  divisors  and  the  last  qitotient  are  the  prime  factors. 


Proof. — TJie  product  of  all  the  prime  factors 
the  given  number.     (Prin".  4.) 


equal  to 


Resolve  the  following  numbers  into  their  prime  factors 

8.  1155.  12.     13981.  16.     12673. 

9.  2934.  13.     32320.  17.     10010. 

10.  6300.  14.     21504.  18.     28665. 

11.  2205.  15.     29925.  19.     31570. 


COMMON     DIVISORS.  83 

COMMON   DIVISORS. 

ORAL   exercises. 

166.  1.  Name  two  exact  divisors  of  12.    Of  15.    Of  20. 

2.  Name  three  exact  divisors  of  24.     Of  48.     Of  72. 

3.  What  number  is  an  exact  divisor  of  27  and  of  56  ? 

'  4.  What  are  the  prime  divisors  of  15  ?    55  ?    49  ?    77? 

5.  What  are  the  composite  divisors  of  72  ?     84  ?     120  ? 

6.  What  prime  divisor  is  common  to  28,  35,  and  42  ? 

7.  Name  a  common  measure  of  22,  44,  and  66. 

8.  Name  the  greatest  common  measure  of  16,  32,  and  64. 

9.  Of  what  three  numbers  is  12  a  common  divisor  ? 

10.  What  two  numbers  will  exactly  divide  15  and  30  ? 
Their  sum  and  difference  ? 

11.  What  is  the  smallest  exact  divisor  of  the  sum  and 
difference  of  10  and  15  ?     Of  21  and  56  ? 

12.  What  is  the  greatest  exact  divisor  of  the  sum  and 
difference  of  16  and  24  ?     Of  18  and  45  ? 

13.  Find  the  greatest  common  measure  of  14, 42,  and  56. 

14.  Find  the  greatest  common  divisor  of  27,  36,  and  45. 

DEFINITIONS    AND    PKINCIPLES. 

167.  A  Common  Divisor  of  two  or  more  numbers 
is  a  common  factor  of  each  of  them. 

168.  The  Greatest  Common  Divisor  of  two  or 

more  numbers  is  the  greatest  common  factor,  and  is  the 
product  of  all  the  common  prime  factors. 

169.  Principles.— 1.  The  only  exact  divisors  of  a 
number  are  its  prime  factors,  or  the  product  of  two  or  more 
of  them. 


84  PROPERTIES     OF     NUMBERS. 

2.  An  exact  divisor  divides  any  number  of  times  xts 
dividend. 

3.  A  common  divisor  of  two  or  more  numbers  will  divide 
their  sum,  and  also  the  difference  of  any  two  of  them. 

4.  The  greatest  common  divisor  of  two  or  more  numbers 
is  the  product  of  all  their  common  prime  factors. 

WRITTEN     EXERCISES. 

170.  When  the  numbers  can  be  readily  factored. 

1.  What  is  the  greatest  common  divisor  of  42,  63,  and  126  ? 
1  st  operation.  Analysis. — By  factoring  the  given  num- 

irt  7v3v9      hers,  the  prime  factors  common  to  all  of 

them  are  7  and  3.     Hence  7  x  3  =  21  is 
bo  =  7  X  o  X  o      the  greatest  common  divisor  of  42,  63,  and 
126  =  7x3x6     126.    (Prin.  4.) 

2d  operation.  Analysis. — Since  the  given  numbers 

3)42       63       126      are  exact-ly  divisible  by  3,  and  the  result 

~ ing  quotients  by  7,  they  are  also  divisible 

7  )14       21  42      by  7  x  3,  or  21.     (Prin.  1.) 

2  3  Q         If  there  were  other  factors  of  the  great- 

est common  divisor,  then  the  quotients  2. 
3,  and  6  would  be  exactly  divisible  by  them. 

Find  the  greatest  common  divisor 


2.  Of  42  and  112. 

3.  Of  96  and  544. 


4.  Of  40,  75,  and  100. 

5.  Of  72,  126,  and  216. 


Rule. — Separate  the  numbers  into  their  prime  factors 
and  find  the  product  of  all  that  are  common.    Or, 

I.  Write  the  numbers  in  a  line,  and  divide  by  any  prime 
factor  common  to  all  the  numbers. 

II.  Divide  the  quotients  in  like  manner,  and  so  continue 
the  division  till  all  the  quotients  are  prime  to  each  other. 

III.  The  product  of  all  the  divisors  ivill  be  the  greatest 
common  divisor.     (Prin.  4. ) 


COMMON     DIVISORS. 


85 


What  is  the  greatest  common  divisor 


6.  Of  144  and  720  ? 

7.  Of  308  and  506? 


8.  Of  126,  210,  and  252? 

9.  Of  72,  96, 120,  and  384? 


171.  When  the  numbers  cannot  be  readily  factored, 

1.  Find  the  greatest  common  divisor  of  527  and  1207. 


527 
459 


OPERATION. 

1207 
1054 


68 
68 


Analysis. — Draw  two  vertical  lines,  and 
place  the  greater  number  on  the  right,  and  the 
less  on  the  left,  one  line  lower  down.  Di- 
vide 1307  by  527,  and  write  the  quotient  2 
between  the  vertical  lines,  the  product,  1054, 
under  the  greater  number,  and  the  remainder 
153,  below. 
Next,  divide  527  by  this  remainder  153, 
writing  the  quotient  3  between  the  verticals,  the  product  459,  on 
the  left,  and  the  remainder  68,  below. 

Again,  divide  the  last  divisor  153,  by  68,  and  write  the  product, 
and  remainder  in  the  same  order  as  before. 

Finally,  dividing  the  last  divisor  68,  by  the  last  remainder  17, 
there  is  no  remainder.  Hence  17,  the  last  divisor,  is  the  greatest 
common  divisor  of  537  and  1207. 

Proof. — Now,  observing  that  the  dividend  is  always  the  sum  of 
the  product  and  remainder,  and  that  the  remainder  is  always  the 
difference  of  the  dividend  and  product,  trace  the  work  in  the  reverse 
order,  as  indicated  by  the  arrow  line  in  the  diagram  below. 

17  divides  68,  as  proved  by  the 
last  division  ;  it  will  also  divide  2 
times  68,  or  136  (Prin.  2).  Since 
17  divides  both  itself  and  136,  it  will 
divide  153,  their  sum  (Prin.  3).  It 
will  also  divide  3  times  153,  or  459 
(Prin.  2)  ;  and,  since  it  is  a  common 
divisor  of  459  and  68,  it  must  divide 
their  sum,  527,  which  is  one  of  the 
given  numbers.  It  will  also  divide 
2  times  527,  or  1054  (Prin.  2) ;  and, 
since  it  divides  1054  and  153,  it  must 
divide  their  sum,  1207,  the  greater  number  (Prin.  3.)  Hence,  17 
is  a  common  divisor  of  the  given  numbers. 


ILLUSTRATION. 

A 

527 

2 

459 

3 

68 

2 

AQ 

4 

1207 


1054 


153 


136 


17 


86 


PROPERTIES     OF     LUMBERS. 


527^ 


459 


68 


Again,  tracing  the  work  in  the  direct  order,  as  indicated  in  the 
ioo7  following  diagram,  the  greatest  com- 
mon divisor,  whatever  it  is,  must 
divide  2  times  527,  or  1054  (Prin.  2). 
And  since  it  will  divide  both  1054 
and  1207,  it  must  divide  their  dif- 
ference, 153  (Prin.  3).  It  will  also 
divide  3  times  153,  or  459  (Prin.  2) ; 
and  as  it  will  divide  both  459  and 
527,  it  must  divide  their  difference, 
68  (Prin.  3).  It  will  also  divide 
2  times  68,  or  136  (Prin.  2) ;  and  as 
it  will  divide  both  136  and  153,  it 

must  divide  their  difference,  17  (Prin.  3) ;  hence,  it  cannot  be  greater 

than  17. 


1054 


153 


136 


17 


Thus,  it  has  been  shown, 

1st.  That  17  is  a  common  divisor  of  the  given  numbers. 
2d.    That  their  greatest  common  divisor,  whatever  it  be,  cannot 
be  greater  than  17.    Hence  it  must  be  17. 

In  like  manner,  find  the  greatest  common  divisor 

6.  Of    825  and  1372. 

7.  Of  2041  and  8476. 

8.  Of  7241  and  10907. 

9.  Of  2373  and  6667. 


2.  Of  316  and  664. 

3.  Of    679  and  1869. 

4.  Of  1080  and  189. 

5.  Of  2192  and  458. 


Rule. — I.  Draw  two  vertical  lines,  and  ivrite  the  two 
numbers,  one  on  each  side,  the  greater  number  one  line 
above  the  less. 

II.  Divide  the  greater  number  by  the  less,  writing  the 
quotient  between  the  verticals,  the  product  under  the  divi- 
dend, and  the  remainder  below. 

III.  Divide  the  less  number  by  the  remainder-,  the  last 
divisor  by  the  last  remainder,  and  so  on,  till  nothing  re- 
mains.    Tlie  last  divisor  is  the  greatest  common  divisor. 


COMMON     DIVISORS.  87 

IV.  If  more  than  two  numbers  are  given,  first  find  the 
greatest  common  divisor  of  two  of  them,  and  then  of  this 
divispr  and  one  of  the  remaining  numbers,  and  so  on  to  the 
last ;  the  last  common  divisor  found  is  the  greatest  common 
divisor  of  all  the  given  numbers. 

10.  What  is  the  greatest  number  that  will  divide  3281 
and  10778  ?     10353  and  14877  ? 

11.  What  is  the  greatest  number  that  will  divide  620, 
1116,  and  1488  ?     396,  5184,  and  6914? 

12.  A  man  having  a  piece  of  land,  the  sides  of  which 
are  240  feet,  648  feet,  and  420  feet,  wishes  to  inclose  it 
with  a  fence  having  panels  of  the  greatest  possible  uni- 
form length  ;  what  will  be  the  length  of  each  panel  ? 

13.  A  farmer  wishes  to  put  231  bushels  of  corn,  393 
bushels  of  wheat,  and  609  bushels  of  oats  into  the  largest 
bags  of  equal  size,  that  will  exactly  hold  each  kind.  How 
many  bushels  must  each  bag  hold  ? 

14.  A  forwarding  merchant  has  15292  bushels  of  wheat, 
1520  bushels  of  corn,  and  504  bushels  of  beans,  which  he 
wishes  to  ship,  in  the  fewest  bags  of  equal  size  that  will 
exactly  hold  either  kind  of  grain  ;  how  many  bags  will 
it  take  ? 

15.  Three  persons  have  respectively  $630,  $1134,  and 
$1386,  with  which  they  agree  to  purchase  horses,  at  the 
highest  price  per  head,  that  will  allow  each  man  to  invest 
all  his  money.     How  many  horses  can  each  man  buy  ? 

16.  How  many  rails  will  inclose  a  field  5850  feet  long 
by  1729  feet  wide,  the  fence  being  straight,  and  7  rails 
high,  and  the  rails  of  equal  length,  and  the  longest  that 
can  be  used  ? 


88  PROPERTIES     OP     NUMBERS. 

MULTIPLES. 

ORAL    EXERCISES.  , 

172.  1.  What  numbers  between  5  and  30  are  exactly 
divisible  by  4  ?    By  6  ?     7  ?     8  ?     9  ? 

2.  What  numbers  less  than  40  are  exactly  divisible  by  7  f 

3.  What  prime  factors  are  common  to  6,  and  5  times  6  ? 

4.  Name  some  numbers  exactly  divisible  by  4  and  6 ; 
by  3  and  7  ;  by  5  and  7  ;  by  8  and  10. 

5.  By  what  three  prime  numbers  can  42  be  divided  ? 

6.  Name  some  numbers  of  which  3  and  4  are  factors. 

7.  Find  the  least  number  exactly  divisible  by  3,  4,  and  5. 

DEFINITIONS    AND    PRINCIPLES. 

173.  A  Multiple  of  a  number  is  a  number  exactly 
divisible  by  the  given  number ;  or,  it  is  any  product  or 
dividend  of  which  a  given  number  is  a  factor. 

1.  A  number  may  have  an  unlimited  number  of  multiples. 

2.  A  number  is  a  dicisor  of  all  its  multiples  and  a  multiple  of  all 
its  divisors. 

174.  A  Common  Multiple  of  two  or  more  given 
numbers  is  a  number  exactly  divisible  by  each  of  them. 

175.  The  Least  Common  Multiple  of  two  or 

more  given  numbers  is  the  least  number  exactly  divisible 
by  each  of  them. 
Two  or  more  numbers  can  have  but  one  least  common  multiple. 

176.  Principles. — 1.  A  multiple  of  a  number  contains 
each  of  the  prime  factors  of  that  number. 

2.  A  common  multiple  of  two  or  more  numbers  contains 
each  of  the  prime  factors  of  those  numbers.     Hence, 


MULTIPLES.  89 

3.  TJie  least  common  multiple  of  two  or  more  numbers  is 
the  least  number  that  contains  each  of  the  prime  factors  of 
those  numbers. 

4.  A  common  multiple  of  two  or  more  numbers  may  be 
found  by  multiplying  the  given  numbers  together. 

WRITTEN     EXERCISES. 

1*77.  To  find  the  least  common  multiple. 

FIRST    METHOD. 

1.  Find  the  least  common  multiple  of  30,  42,  and  66. 

operation.  Analysis. — The  least  common 

,     oa  9  y  3  y-  5  multiple  cannot  be  less  than  the 

largest  number  66,  since  it  must 
42=2x3X7  contain   66  ;   hence  it  must  con- 

66=2x3x11  tain  all  the  prime  factors  of  66, 

2x3x11x7x5  =  2310     ^  af  2-  3-  and  "•  ^«  M 

But  the  least  common  multiple  of 
66  must  also  contain  all  the  prime  factors  of  each  of  the  other  num- 
bers, and  since  the  prime  factors  2  and  3  of  GG  are  common  also  to 
42  and  30  omit  them,  and  annex  the  factors  7  and  5  to  those  of  66, 
and  the  series  2,  3,  11,  7,  and  5  are  all  the  prime  factors  of  the 
given  numbers,  and  their  product  2  x  3  x  11  x  7  x  5  =  2310,  is  the 
least  common  multiple  of  the  given  numbers.     (Prin.  3.) 

2.  Find  the  least  common  multiple  of  24,  42,  and  17. 

3.  Find  the  least  common  multiple  of  8, 12,  20,  and  30. 

4.  Find  the  least  common  multiple  of  10, 45,  75,  and  90. 

Rule. — I.  Resolve  each  of  the  given  numbers  into  its 
prime  factors. 

II.  Multiply  together  all  the  prime  factors  of  the  largest 
number,  and  such  prime  factors  of  the  other  numbers  as 
are  not  found  in  the  largest  number,  and  their  product 
will  be  the  least  common  multiple. 


90  PROPERTIES     OF     NUMBERS 

Find  the  least  common  multiple 


5.  Of  30,  66,  78,  and  42. 

6.  Of  21,  30,  44,  and  126. 


7.  Of  16,  60,  140,  and  210. 

8.  Of  16,  48,  80,  32,  and  66. 


SECOND    METHOD. 

1 78. 1.  Find  the  least  common  multiple  of  18, 24,  and  54. 

operation.  Analysis. — Write  the  numbers  in  a  hori- 

2    18        2  4        5  4     zontal  line,  with  a  vertical  line  at  the  left. 
Since  2  is  a  prime  factor  of  one  or  more 
of  the  given  numbers,  it  must  also  be  a 


1  8 

24 

54 

9 

12 

27 

3 

4 

9 

4 

3 

3        3  4  9      factor  of  the  least  common  multiple  of 

those  numbers.  (Prin.  3.)  Hence,  divide 
by  2  and  write  the  quotients  underneath. 
For  a  like  reason  divide  again  successively  by  3  and  3,  writing  the 
quotients  and  undivided  numbers  in  a  line  below,  omitting  to  write 
any  quotient  when  it  is  1. 

Since  there  is  no  factor  common  to  4  and  3,  they  are  prime  to 
each  other,  and  hence  the  divisors  2,  3,  and  3,  with  the  numbers  4 
and  3  in  the  last  line,  are  all  the  prime  factors  of  the  given  numbers, 
#nd  their  product  21G  is  the  least  common  multiple.     (Prin.  3.) 

If  in  any  example,  any  of  the  smaller  numbers  are  exactly  con- 
tained in  the  larger,  they  may  be  omitted  in  finding  the  least  com- 
mon multiple,  inasmuch  as  a  number  that  will  contain  a  given 
number,  will  contain  any  factor  of  that  number. 

Thus,  if  required  to  find  the  least  common  multiple  of  8,  12,  24, 
72,  and  120,  omit  all  the  numbers  except  72  and  120,  since  the  others 
are  factors  of  these,  and  the  least  common  multiple  of  72  and  120, 
will  be  the  least  common  multiple  of  all  the  numbers. 

2.  Find  the  least  common  multiple  of  32,  34,  and  36. 

3.  Find  the  least  common  multiple  of  84, 100,  and  224. 

Rule. — I.  Write  the  numbers  in  a  horizontal  line,  omit- 
ting such  of  the  smaller  numbers  as  are  factors  of  the 
larger,  and  draw  a  vertical  line  at  the  left. 

II.  Divide  by  any  prime  factor  that  will  exactly  divide 
two  or  more  of  the  given  numbers,  and  write  the  quotients 
and  undivided  numbers  in  a  line  underneath. 


MULTIPLES.  .  91 

III.  In  like  manner  divide  the  quotients  and  undivided 
numbers  until  they  are  prime  to  each  other. 

IV.  The  product  of  the  divisors  and  the  final  quotients 
and  undivided  numbers,  is  the  least  common  multiple. 

What  is  the  least  common  multiple 


4.  Of  4662,  and  5698  ? 

5.  Of  312,  260,  and  390  ? 


6.  Of  24, 10,  32,  45  and  25  ? 

7.  Of  153, 204, 102,  and  1020? 


8.  Find  the  least  common  multiple  of  the  first  eight 
even  numbers. 

9.  Find  the  least  common  multiple  of  the  first  five 
odd  numbers. 

10.  What  is  the  least  number  of  oranges  that  can  be 
equally  distributed  among  16,  20,  24,  or  30  boys  ? 

11.  What  is  the  shortest  piece  of  rope  that  can  be  cut 
exactly  into  pieces  either  15,  18,  or  20  feet  long  ? 

12.  What  is  the  smallest  sum  of  money  which  can  be 
exactly  expended  for  books  at  $5,  or  $3,  or  $4,  or  $6 
each  ? 

13.  What  is  the  product  of  the  least  common  multiple 
of  12,  16,  24,  and  32,  multiplied  by  their  greatest  common 
divisor  ? 

14.  Divide  the  least  common  multiple  of  7,  42,  6,  9, 
10,  and  630,  by  the  greatest  common  divisor  of  110,  140, 
and  680. 

15.  What  is  the  smallest  sum  of  money  which  can  be 
exactly  expended  for  sheep  at  $8,  or  cows  at  $28,  or  oxen 
at  $54,  or  horses  at  $162  each  ? 

16.  What  is  the  smallest  quantity  of  grain  that  will  fill 
an  exact  number  of  bins,  whether  they  hold  36,  48,  80,  or 
144  bushels  ? 


92  PEOPEBTIES    OF     NUMBERS. 


CANCELLATION. 

ORAI      EXERCISES. 

179.  .1.  Divide  72  by  24.  One-half  of  72  by  one-half 
of  24.     One-third  of  72  by  one-third  of  24. 

2.  Divide  one-fourth  of  72  by  one-eighth  of  24. 

3.  Divide  36  by  9.     One-third  of  36  by  one-third  of  9. 

4.  What  factors  are  common  to  72  and  24  ? 

5.  What  is  the  quotient  of  12  x  6  divided  by  12x2? 

6.  Divide  3x3x3by  3x3.     4  x5  x2  by  2x2  x5. 

7.  Divide7x6x2by2x6x7.     5  x6  x4  by  3  x4X6. 

DEFINITIONS    AND    PEINCIPLES. 

180.  Cancellation  is  the  process  of  abridging 
operations  in  division  by  rejecting  equal  factors  from  both 
dividend  and  divisor. 

181.  Pkinciples. — 1.  Rejecting  a  factor  from  any 
number  divides  the  number  by  that  factor. 

2.  Rejecting  equal  factors  from  both  dividend  and  divi- 
sor does  not  change  the  quotient. 

WRITTEN     EXERCISES. 

182.  1.  Divide  56  x  24,  by  48  X  7. 

1st  operation.  Analysis.— Indicate  tlie 

56x24        $X#X0X4_         operation  to  be  performed 

4g  x  y  =       $~X^$  X  f       ~~         in  tlie  examPle>  DV  writing 

the  numbers  that  constitute 
the  dividend,  above  a  line,  and  those  that  constitute  the  divisor, 
below  it. 

Resolve  these  numbers  into  their  factors,  and  the  dividend  will 
consist  of  8  x  7  x  6  x  4,  and  the  divisor  of  8  x  6  x  7.  Rejecting  equal 
factors  from  both  dividend  and  divisor,  there  remains  the  factor  4 
in  the  dividend.     Hence  the  quotient  is  4. 


CANCELLATION 


93 


2d  opekation. 

3f         4 
00  X  t4 


4$  x  t 


Analysis. — Since  it  is  evident  that  8  will 
divide  both  56  and  48,  reject  8  as  a  factor  of 
56,  retaining  the  factor  7,  and  also  of  48,  re- 
=  4        taining  the  factor  6. 

Again,  since  6  will  divide  both  24  in  the 
dividend  and  6  in  the  divisor,  reject  6  as  a  fac- 
tor from  both,  retaining  the  factor  4  in  the  dividend.  Finally, 
rejecting  the  factor  7,  common  both  to  the  dividend  and  to  the 
divisor,  there  remains  only  the  factor  4  in  the  dividend,  which  is 
the  required  quotient. 

2.  Divide  the   product  of  44,  30,  7,  and  6,  by  the 
product  of  33,  18,  and  14  ;  or,  divide  55440  by  8316. 


55440 
8316 


OPERATION. 
2         10 

44x$0xflx$ 

$$x*$x*4  '' 
3       $        % 


Or, 


10x2 


=  6f 


44^ 
*0lO 

t 

0 

3 

20 

6f 

By  many  it  is  thought  more  convenient  to 
write  the  factors  of  the  dividend  on  the  right  of  a 
vertical  line,  and  the  factors  of  the  divisor  on  the 
left. 

3.  Divide  13x7x5x3  by  3x5x7. 

4.  Divide  42  x  18  x  6  x  4  by  36  x  21  x  6. 


Kule. — I.  Cancel  all  the  factors  common  to  doth  divi- 
dend and  divisor. 

II.  Divide  the  product  of  the  remaining  factors  of  the 
dividend  hy  the  product  of  the  remaining  factors  of  the 
divisor,  and  the  result  will  be  the  quotient. 

When  a  factor  equal  to  the  number  itself  is  canceled,  the  unit  1 
remains,  since  a  number  divided  by  itself  gives  a  quotient  of  1 .  If 
the  1  occur  in  the  dividend,  it  must  be  retained ;  if  in  the  divisor, 
it  need  not  be  regarded. 


94  PKOPERTIES     OF     NUMBERS. 

5.  What  is  the  quotient  of  35  x  33  x  28,  divided  by  15  x 
14x11? 

6.  What  is  the  quotient  of  140  x  39  x  13  x  7,  divided  by 
7x26x21? 

7.  Multiply  11  times  21  by  26,  and  divide  the  product 
by  14  times  13. 

8.  How  many  times  is  the  continued  product  of  14,  9, 
3,  20,  5,  and  6  contained  in  the  continued  product  of 
183,  18,  70,  12,  and  5  ? 

9.  If  213  x  190  x  84  x  264  is  the  dividend,  and  56  times 
36  multiplied  by  30  is  the  divisor,  what  is  the  quotient  ? 

10.  (240x56x18) ^-(60x28x9)=? 

11.  (72x48x28x5)-^(84xl5x7x6)=? 

12.  (66x18x27x25)^.(84x45x7x30)=? 

13.  (80  x  60  x  50  x  16  x  14)  -f- (70  x  50  x  24  x  20)  =  ? 

14.  Multiply  64  by  7  times  31,  divide  the  product  by 
8  times  56,  multiply  this  quotient  by  15  times  88,  divide 
the  product  by  55,  multiply  this  quotient  by  13,  and  di- 
vide the  product  by  4  times  6.     What  is  the  quotient  ? 

-,K    m  ix,  x.     A    *  12x60x27x35 

15.  Find  the  quotient  of 


16.  Find  the  quotient  of 


7x15x42x108 
77  x  100  x  18  x  64 


25x11x49x16 

17.  How  many  tons  of  hay  at  $18,  must  be  given  for 
45  cords  of  wood  at  $4  a  cord  ? 

18.  How  many  flour  barrels  at  $.80  each,  will  pay  for 
112  bushels  of  corn  at  $.70  a  bushel  ? 

19.  How  many  tubs  of  butter,  each  containing  56 
pounds,  at  30  cents  a  pound,  must  be  given  for  7  barrels 
of  sugar,  each  containing  195  pounds,  at  10  cents  a  pound  ? 


CANCELLATION.  95 

20.  A  laborer  gave  12  days'  work  for  48  busnels  of 
potatoes,  worth  50  cents  a  bushel.  What  were  his  daily 
earnings  ? 

21.  A  grocer  sold  24  boxes  of  soap,  each  containing  55 
pounds,  at  10  cents  a  pound,  and  received  as  pay  88  bar- 
rels of  apples,  each  containing  3  bushels.  How  much 
were  the  apples  worth  a  bushel  ? 

22.  Sold  20  pounds  of  butter  at  27  cents  a  pound,  which 
exactly  paid  for  15  pounds  of  coffee.  What  was  the  price 
of  the  coffee  a  pound  ? 

23.  A  farmer  exchanged  240  bushels  of  corn,  worth  $.75 
a  bushel,  for  an  equal  number  of  bushels  of  barley,  worth 
$1  a  bushel,  and  oats,  worth  $.50  a  bushel.  How  many 
bushels  of  each  did  he  receive  ? 

24.  A  farmer  bought  two  kinds  of  cloth,  one  kind  at 
$.75  a  yard,  and  the  other  at  $.90,  buying  twice  as  many 
yards  of  the  first  kind  as  of  the  second.  He  paid  for  the 
cloth,  132  pounds  of  butter  at  40  cents  a  pound.  How 
many  yards  of  each  kind  of  cloth  did  he  buy  ? 

25.  A  merchant  bought  6  loads  of  oats,  each  load  con- 
taining 22  bags,  and  each  bag  2  bushels,  worth  56  cents 
a  bushel.  He  gave  in  payment  8  boxes  of  tea,  each  con- 
taining 24  pounds.     What  was  the  tea  worth  a  pound? 

26.  How  many  bushels  of  oats  at  $.60  a  bushel,  will 
pay  for  12  tons  of  coal  at  $7.20  a  ton? 

27.  How  many  chests  of  tea,  each  containing  63  pounds, 
worth  87-J-  cents  a  pound,  must  be  given  for  21  bags  of 
coffee,  each  weighing  28  pounds,  worth  37£  cents  a 
pound  ? 

28.  How  many  days'  work,  at  $1.25  a  day,  will  pay  for 
75  bushels  of  corn,  at  $.80  a  bushel  ? 


96 


PROPERTIES     OF     NUMBERS. 


183. 


SYNOPSIS  FOR  REVIEW. 


1.  Definitions. 


Divisibility  op 
Numbers. 


3.  Factoring. 


1.  Properties  of  Numbers.  2.  Inte- 
gral Number,  or  Integer.  3.  Even 
Number.  4.  Odd  Number. 
5.  Prime  Number.  6.  Composite 
Number.  7.  Factors.  8.  Prime 
Factor.    9.  Exact  Divisor. 

1.  How  to  find  whether  a  number  is 
divisible  by  2,  3,  4,  5,  6,  8,  9,  or 
10  ;  also  by  7,  11,  or  13. 

2.  Other  properties  of  even  and  of 
prime  numbers. 


1.  Definitions. 


j  1.  Factoring. 
\  2. 


4.  Common  Divisors.  < 


5.  Multiples. 


6.  Cancellation. 


Exponent. 
2.  Principles,  1,  2,  3,  4. 
L  3.  Rule.  4.  Proof. 

f  1.  Divisof. 

1.  Definitions.  <  2.  Common  Divisor. 

[  3.  O.  C.  Divisor. 

2.  Principles,  1,  2,  3,  4. 

3.  Rule  (1st),  I,  II,  III. 

4.  Rule  (2d),  I,  II,  III,  IV. 

f  1.  Multiple. 

1.  Definitions.  1  2.  G.  Multiple. 

[  3.  L.  G.  Multiple. 

2.  Principles,  1,  2,  3. 

3.  Rule  (1st),  I,  II. 

4.  Rule  (2d),  I,  II,  III,  IV. 

1.  Definition. 

2.  Principles,  1,  2. 

3.  Rule,  I,  II. 


IMM^MSj^MIIliiifu 


OJRJLZ     EXERCISES. 

184.  1.  If  any  unit,  as  an  apple,  or  a  yard,  be  divided 
into  2  equal  parts,  what  is  each  part  named  ?     One-half. 

2.  If  the  unit  be  divided  into  3  equal  parts,  what  name 
is  given  to  1  of  the  parts  ?    To  2  of  the  parts  ? 

3.  If  the  unit  be  divided  into  5  equal  parts,  what  is  each 
part  named  ?    What  name  is  given  to  3  of  the  parts  ? 

4.  How  many  halves  are  there  in  a  unit  ?    How  many 
thirds?    Fourths?    Fifths?    Sixths?    Sevenths? 

5.  If  a  mile  be  divided  into  4  equal  parts,  what  part  of 
the  whole  mile  is  1  of  the  parts  ?    3  of  the  parts  ? 

6.  What  is  1  of  5  equal  parts  of  a  unit  called  ?  What 
are  2  of  6  equal  parts  called  ?     4  of  10  equal  parts  ? 

7.  What  is  meant  by  1  sixth  of  a  unit  ?    By  3  fourths  ? 

8.  What  are  3  of  the  7  equal  parts  of  a  week  called  ? 

9.  Which  is  the  smaller,  one-third  or  one-fourth  ?  One- 
fifth  or  one-third  ? 

10.  Which  is  the  greater,  one-fourth  or  one-sixth  ? 

185.  Principles.— 1.  Tlie  less  the  number  of  equal 
parts  into  which  a  unit  is  divided,  the  greater  is  the 
value  of  each  part. 

2.  The  greater  the  number  of  equal  parts  into  which 
a  unit  is  divided,  the  less  is  the  value  of  each  part. 
5 


98  FRACTIONS. 

DEFINITIONS. 

186.  A  Fraction  is  one  or  more  of  the  equal  parts 
of  a  unit.     Thus,  1  half  and  2  thirds  are  fractions. 

187.  A  Fractional  Unit  is  one  of  the  equal  parts 
into  which  any  unit  is  divided.  Thus,  1  fourth  and 
1  fifth  are  fractional  units  of  fourths  and  fifths. 

Fractional  units  take  their  name  and  their  value  from  the  number 
of  parts  into  which  the  integral  unit  is  divided. 

188.  A  fraction  is  usually  expressed  hy  two  numbers, 
called  the  Numerator  and  the  Denominator,  one  written 
over  the  other  with  a  line  between  them.  A  fraction 
written  in  this  form  is  sometimes  called  a  Common  Frac- 
tion.    Thus, 


One-third  is  written  \ 
Three-fourths  "  £ 
Five-sixths  "       f 


Nine-tenths  is  written  -fa 

Seven-twentieths  "      -fo 

Twelve-thirty-fifths        "       ff 
Seven-eighths      u       -J    Thirty-six  forty-ninths    "       fjj- 

189.  The  Denominator  of  a  fraction  shows  the 

number  of  equal  parts  into  which  the  unit  is  divided,  and 

also  indicates  the  name  of  these  parts.    It  is  written  below 

the  line. 

Thus,  in  the  fraction  f ,  8  is  the  denominator  and  shows  that  the 
unit  is  divided  into  eight  equal  parts,  named  eighths. 

190.  The  Numerator  of  a  fraction  shows  the  num- 
ber of  equal  parts  taken  to  form  the  fraction.  It  is 
written  above  the  line. 

Thus,  in  f ,  7  is  the  numerator,  and  shows  that  7  of  the  8  equal 
parts  are  taken,  or  expressed  by  the  fraction.* 

191.  The  Terms  of  a  fraction  are  its  numerator  and 
denominator.   Thus,  6  and  7  are  the  terms  of  the  fraction  £. 


FKACTIONS.  99 


Express  by  figures, 

1.  Five-ninths. 

2.  Seven  twenty-fifths. 

3.  Nine-eighteenths. 

4.  Twelve  twentieths. 


6.  Twenty-six  forty-eighths. 

7.  Twenty-seven  two-hundredths. 

8.  Forty-three  ninety-ninths. 

9.  Sixteen  one-hundred-eighths. 


5.  Eight  thirty-sixths.    10.  Fifty-five  eighty-ninths. 
Copy  and  read, 

**•    "^B"  J      A"  5  TE'V  ?450?      TTF  5      TOT  j      foo  • 

192.  Fractions  are  Proper  or  Improper. 

193.  A  Proper  Fraction  is  a  fraction  whose  nu- 
merator is  less  than  its  denominator.  Its  v«/we  is  less 
than  a  unit.    Thus,  ^,  -f,  and  \^  are  proper  fractions. 

194.  An  Improper  Fraction  is  a  fraction  whose 
numerator  equals  or  exceeds  its  denominator.  Its  value 
is  equal  to,  or  greater  than  a  unit.  Thus,  -£,  J^,  and  *£■ 
are  improper  fractions. 

195.  A  Mixed  Number  is  an  integer  and  a  fraction 
united.     Thus,  12f  is  equivalent  to  12  +  $. 

196.  The  Reciprocal  of  a  number  is  1  divided  by 
that  number.  Thus,  the  reciprocal  of  9  is  1-^-9=^;  of 
16,  it  is  1  -~  16  =  -fa,  etc. 

197.  The  Reciprocal  of  a  Fraction  \z  1  divided 
by  that  fraction,  or  it  is  the  fraction  inverted.  Thus,  the 
reciprocal  of  f  is  1  -f- £  =  £  ;  of  -^,  it  is  -^. 

198.  The  Value  of  a  fraction  is  the  quotient  of  its 
numerator  divided  by  its  denominator.     Thus,  ^  =  4. 


100  FRACTION'S. 

1.  Analyze  the  fraction  -J. 

Analysis. — |  is  a  fraction  ;  8  is  the  denominator,  and  shows 
that  the  unit  is  divided  into  8  equal  parts  ;  \  is  the  fractional  unit, 
since  it  is  one  of  the  eight  equal  parts  into  which  the  unit  is  divided  ; 
■7  is  the  numerator,  and  shows  that  seven  of  these  equal  parts  are 
taken  ;  7  and  8  are  the  terms  of  the  fraction.  It  is  a  proper  frac- 
tion, since  the  numerator  is  less  than  the  denominator ;  its  value  is 
less  than  1 ;  and  it  is  read  seven-eighths. 

In  like  manner,  analyze 

2.  f.     |     3.    H.     |     4.    f.     f-   5.    «.;  |     G.    ^. 

199.  Since  fractions  indicate  division,  all  changes  in 
the  terms  of  a  fraction  will  affect  the  value  of  the  fraction 
according  to  the  laws  of  division  ;  hence  if  we  substitute 
the  General  Principles  of  Division  (142),  we  shall  have 
the  following 

200.  GENERAL  PRINCIPLES  OF  FRACTIONS. 

1.  Multiplying  the  numerator,  or  )  , ,  7 . .  , .     ,x   , 

«.  ./.      T    -,         •     ,  Y  Multiplies  the  fraction. 

Dividing  the  denominator,         ) 

2.  Dividing  the  numerator,  or       )  ^.^  tufmctim. 
Multiplying  the  denominator,   ) 

3.  Multiplying  or  dividing  both  J  ^  ^   cJi  (he 

numerator  and  denominator  V     ^  ofihefrae(im. 
by  the  same  number,  ) 

201.  These  three  principles  may  be  embraced  in  one 

GENERAL    LAW. 

A  change  in  the  numerator  produces  a  like  change  in 
the  value  of  the  fraction  ;  but  a  change  in  the  denomina  tor 
produces  an  opposite  change  in  the  value  of  the  fraction. 


REDUCTION.  101 

EEDUCTIOK      . 

202.  To  reduce  fractions  to  higher  or  lower  terms. 
ORAL     exercises. 

1.  One-half  is  equal  to  how  many  fourths  ? 

Analysis. — Since  1  is  equal  to  4  fourths,  ^  is  equal  to  1  half  of  4 
fourths  or  2  fourths. 

2.  One-third  of  a  mile  is  how  many  sixths  of  a  mile  ? 

3.  One-half  of  a  dollar  is  how  many  fourths  of  a  dollar  ? 

4.  Name  some  equivalent  fractions  for  halves.     Thirds. 

5.  Express  f  in  terms  3  times  as  great.  4  times  as  great. 

6.  The  denominators  four,  six,  eight,  and  ten,  are  mul- 
tiples of  what  number? 

7.  Multiply  both  terms  of  |  by  3,  and  show  that  the 
value  of  the  fraction  is  not  changed. 

Analysis. — If  both  terms  of  £  are  multiplied  by  3,  the  resulting 
fraction  is  f^,  which  is  equivalent  to  |,  since  the  fractional  unit  is 
i  as  great,  while  the  number  taken  is  3  times  as  great. 

8.  Name  three  equivalent  fractions  for  %  ;  for  -J  ;  for  -§. 

9.  Change  -J  to  twelfths.     To  eighteenths. 

10.  8  tivelfths  are  how  many  thirds? 

Analysis. — Since  1  third  is  equal  to  4  twelfths,  8  twelfths  are 
equal  to  as  many  thirds  as  4  twelfths  are  contained  times  in  8 
twelfths,  which  is  2  times.    Hence  there  are  f  in  T\. 

11.  How  many  fourths  of  a  rod  are  9  tivelfths  of  a  rod  ? 

12.  Divide  both  terms  of  ££  by  5,  and  show  that  the 
value  of  the  fraction  is  not  changed. 

Analysis. — If  both  terms  of  ^|  are  divided  by  5,  the  resulting 
fraction  is  f ,  which  is  equivalent  to  -|^,  since  the  fractional  unit  is 
5  times  as  great,  while  the  number  taken  is  ^  as  great. 


102  FEACTIONS. 

13.  Change  |f  to  an  equivalent  fraction  having  a  de 
nominator  1  half  as  great.     1  third  as  great. 

14.  Change  -^  to  a  fraction  having  lower  terms.  £f.  f|. 

15.  In  what  lower  terms  can  |-£  be  expressed  ? 

16.  Change  ^  to  its  lowest  terms.     ££.    £§.     -Ji.     ||. 

17.  Name  two  common  divisors  of  -^-f .    ff.     |-|.    |^. 

18.  Express  T8^  in  terms  4  times  as  great. 

19.  Express  |§  in  terms  6  times  as  great. 


DEFINITIONS. 

203.  Reduction  of  Fractions  is  the  process  of 
changing  their  form  without  altering  their  value. 

204.  A  fraction  is  reduced  to  Higher  Terms  when 
the  numerator  and  denominator  are  expressed  in  larger 
numbers.     Thus,  f  =  -£,  or  ^-. 

205.  A  fraction  is  reduced  to  Lower  Terms  when 
the  numerator  and  denominator  are  expressed  in  smaller 
numbers.     Thus,  -^  =  £,  or  f . 

206.  A  fraction  is  reduced  to  its  Loivest  Terms 

when  its  numerator  and  denominator  are  prime  to  each 
other.     Thus, -^  =  1;  «  =  * 

207.  Fractions  are  changed  to  higher  terms  by  Multi- 
plication, and  to  lower  terms  by  Division. 

All  higher  terms  of  a  fraction  are  multiples  of  its  lowest  terms. 

208.  Peinciple. — Multiplying  or  dividing  both  terms 
of  a  fraction  by  the  same  number  does  not  change  the  value 
of  the  fraction.     (300,  3.) 


REDUCTION.  103 

WRITTEN   EXERCISES. 

209.  1.  Change  £  to  a  fraction  whose  denominator 
is  30. 

operation.  Analysis. — First,  divide  30,  the  required  de- 

3  0  —  6  =  5  nominator,  by  6,  the  denominator  of  the  given 

fraction.     The  quotient  5  is  the  factor  employed 

f  x  5  =  M  to  produce  the  required  denominator.    Hence, 

multiply  both  terms  of  f  by  5  (200,  3),  and  §§ 

is  the  required  fraction. 

2.  Change  ^  to  a  fraction  whose  denominator  is  96. 

3.  Change  f  \  to  a  fraction  whose  denominator  is  105. 

4.  Eeduce  ^q  to  its  lowest  terms. 

operation.  Analysis.— Dividing  both  terms 

72  -^8-—    9.IL-3-— 4         of  the  given  fraction  ■££$,  by  8, 

rro  +  8      tfjh^s     t      (200>  3)  the  result  is  ^    Againj 

Or,  -&qX  It  =  f  dividing  both  terms  of  ■&  by  3, 

the  result  is  f .     Since  the  terms 
of  §  are  prime  to  each  other,  the  lowest  terms  of  ^  are  f . 

The  same  result  is  obtained  more  directly,  by  dividing  both 
terms  by  their  greatest  common  divisor,  24. 

5.  Reduce  ff  to  its  lowest  terms. 

6.  Reduce  ■££%  to  its  lowest  terms. 

7.  Reduce  ££§  to  its  lowest  terms. 

Rules. — 1.  To  reduce  a  fraction. to  higher  terms. 

Divide  the  required  denominator  by  the  denominator  of 
the  given  fraction,  and  multiply  the  terms  of  the  given 
fraction  by  the  quotient. 

2. — To  reduce  a  fraction  to  its  lowest  terms. 

Reject  all  factors  common  to  the  terms  of  the  given  frac- 
tion.   Or, 

Divide  the  terms  of  the  given  fraction  by  their  greatest 
common  divisor. 


104 


FRACTIONS 


8.  Change  T\  to  a  fraction  whose  denominator  is  180. 

9.  Reduce  J  and  -fa  each  to  sixty-thirds. 

10.  Reduce  ■$,  J,  and  -fa,  each  to  120ths. 

11.  Reduce  -f?-,  -J-,  £§>  an^  fi>  eacn  to  132ds. 

12.  Change  168  -±-  252  to  the  form  of  a  fraction  in  its 
lowest  terms.     81  -=-  63.     160  -+-  400.     324  -i-  612. 

Reduce  to  their  lowest  terms, 


13. 

14. 
15. 
16. 


441 
ttt' 


17. 

ttA* 

18. 

tWs- 

19. 

tAt* 

20. 

^JM* 

21. 

22. 
23. 
24. 


5  6  4  3 
5940* 

Httf 

T«ffT- 


210.  To  reduce  an  integer  or  a  mixed  number 
to  an  improper  fraction. 

OBAZ    EXERCISES. 

1.  In  3  units,  how  many  fourths  ? 

Analysis. — Since  in  1  unit  there  are  4fourt7is,  in  3  units  there 
are  3  times  4  fourtns,  or  12  fourths.    Hence  3  =  *£. 

2.  In  4  bushels,  how  many  eighths  of  a  bushel  ? 

3.  How  many  sevenths  of  a  week  in  6  weeks  ? 

4.  How  many  9ths  in  5  ?     6?     8?     10?     12? 

5.  How  many  tenths  of  a  dollar  in  $7  ?    In  $9  ? 

6.  How  many  half  dollars  will  pay  for  a  ton  of  coal 
that  cost  $7  ?    For  a  barrel  of  flour  that  cost  $10  ? 

7.  How  may  an  integer  be  changed  to  thirds?    To 
sixths  f    To  eighths  ?    To  tenths  ? 

8.  In  5-f  how  many  eighths  ? 

Analysis.— Since  1  is  equal  to  8  eighths,  5  equals  5  times  8  eighths, 
or  40  eighths,  and  §  added  make  43  eighths.     Hence  5-|  =  \3-. 

9.  In  6f  cords  of  wood,  how  many  fourths  of  a  cord  ? 


RE  DUCT  I  OK.  105 

10.  How  many  6ths  in  $8|  ?     In  12£  rods  ? 

11.  Among  how  many  boys  can  yon  distribute  5j 
quarts  of  chestnuts,  if  you  give  \  of  a  quart  to  each  ? 

12.  Among  how  many  poor  families  can  4f  tons  of 
coal  be  distributed,  if  each  family  receive  -J-  of  a  ton  ? 

WRITTEN    EXERCISES. 

211.  1.  Change  75  to  the  form  of  a  fraction  having  27 
for  its  denominator. 

operation.  Analysis.  —  Since  1  is  equal  to  27  twenty- 

75  X  27 2025        sevenths,  75  is  equal  to  75 times 27 twenty- 
sevenths,  or  2025  twenty-sevenths.    Hence 
75— ^i.  75  =  -2-ffA 

2.  Change  49^  to  twelfths. 

operation.  Analysis. — Since  1  is  equal  to  12  twelfths, 

49  JL.  49    is  equal  to  49  times  12  twelfths,  or  588 

.j  ~    '  twelfths  ;  to  which  add  ^,  and  the  result  is 

— --  595  twelfths.     Hence  49  £>  =  *&. 

5  88  twelfths  An  integer  is  reduced  to  a  fractional  form 

by  writing  1  under   it   for  a  denominator. 

W  +  T2  =="%-  Thus,  9  =  f ;  23  =  -2^. 

3.  Change  81  to  a  fraction  having  24  for  its  denominator. 

4.  In  78  pounds,  how  many  sixteenths  of  a  pound  ? 

5.  In  42f  weeks,  how  many  sevenths  of  a  week  ? 

6.  Howmany20thsof  aton  in  16^-tons?  In21f£tons? 

Rule. — Multiply  the  integer  hy  the  required  denomina- 
tor, and  to  the  product,  add  the  numerator  of  the  fraction, 
and  under  the  result  tvrite  the  required  denominator. 

Reduce 

7.  207  to  fifteenths.  10.  543^  to  fortieths. 

8.  136f|  to  eighteenths.  11.  184ff  to  ninety-fifths. 

9.  472  jfa  to  twenty-sixths.  12.  2014fj-  to  eighty-fourths. 


106  FRACTIONS. 

13.  Eeduce  204JJ  days  to  twenty-fourths  of  a  day. 

14.  Change  312  to  a  fraction  whose  denominator  is  126. 

15.  Eeduce  2146T3T  to  an  improper  fraction. 

16.  Change  1006-J~£  to  an  improper  fraction. 

212.  To  reduce  an  improper  fraction  to  an  inte- 
ger, or  a  mixed  number. 

OMJLI      EXERCISES. 

1.  How  many  units  are  -^  ? 

Analysis. — Since  4  fourths  equal  1, 18  fourths  are  as  many  times 
1  as  4  fourths  are  contained  times  in  18  fourths,  which  is  4|  times. 

2.  How  many  times  1  are  *f  ?    *$■  ?    ff  ?    \°-  ?    f|  ? 

3.  How  many  yards  are  ^  of  a  yard  ?    ^l  ?    _<y.  ? 

4.  How  many  dollars  are  $i^?     $^?    $^  ?     Iff? 

5.  In  f £  of  a  foot,  how  many  feet  ?    In  -i-p-  of  an  acre, 
how  many  acres  ?     In  J^-  of  a  ton,  how  many  tons  ? 

WRITTEN     EXERCISES. 

213.  1.  Reduce  -S-f8-  to  a  mixed  number. 

operation.  Analysis.— Since  9  ninths  equal  1, 

•     .        _     „        „        rtJ„      218  ninths  are  24f  times  9  nintha 
^4  =  218-9  =  241    HenceV  =  34f, 

2.  Change  ^j8^-  to  a  mixed  number. 

3.  In  *$>  of  a  dollar,  how  many  dollars  ? 

4.  How  many  rods  in  *£$-  of  a  rod  ? 

Rule. — Divide  tlie  numerator  by  the  denominator. 
Reduce  to  integers  or  mixed  numbers, 


5-     W- 
7.     -W- 


8.    m*> 

9.      A^L. 

io.    *$«*. 


n.      ifff*. 

is.     -HMF- 
13.     »w,y>«. 


REDUCTION.  107 

214.  To  reduce  fractions  to  equivalent  fractions 
having  a  common  denominator. 

ORAL      EXERCISES. 

1.  How  many  fourths  in  1  ?    In  J? 

2.  How  many  ninths  in  1  ?    In  £  ?    In  f  ? 

3.  Express  § ,  £,  and  f,  each  as  twelfths. 

4.  Change  f  and  -|  to  fractions  of  the  same  denominator. 

5.  What  is  a  multiple  of  4  ?    Of  6?     Of  8?     Of  9? 

G.  What  is  a  common  multiple  of  3  and  4  ?    Of  4  and  5  ? 

7.  What  is  the  least  common  multiple  of  3,  4,  and  6  ? 

8.  What  is  the  least  common  multiple  of  the  denomi- 
nators of  \,  },  and  £  ?     Of  |,  f,  and  j-? 

9.  Reduce  f  and  \  to  eighteenths.    To  twenty-sevenths. 

10.  Name  some  fractions  that  can  be  changed  to  16ths. 

11.  Name  four  fractions  that  can  be  changed  to  24ths. 

DEFINITIONS    AND    PRINCIPLES. 

215.  A  Common  Denominator  is  a  denomina- 
tor common  to  two  or  more  fractions. 

216.  The  Least  Common  Denominator  of  two 

or  more  fractions  is  the  least  denominator  to  which  they 
can  all  be  reduced. 

Since  all  higher  terms  of  a  fraction  are  multiples  of  its 
corresponding  lowest  terms  (207,  Note),  hence  the  fol- 
lowing 

217.  Principles. — 1.  A  common  denominator  of  two  or 
more  fractions  is  a  common  multiple  of  their  denominators. 

2.  The  least  common  denominator  of  tiuo  or  more  frao 
tions  is  the  least  common  multiple  of  their  denominators. 


108  FRACTIONS, 


.    WRITTEN      EXERCISES. 

218.  1.  Eeduce  \,  f,  and  f  to  equivalent  fractions 
having  a  common  denominator. 

operation.  Analysis. — Multiply  each  denominator  by  the 

n      o      ~      oa      other  two,  and  the  product,  80,  is  a  common  de- 
5iXoXO  =  dO  .  ...     ..*  ' 

nominator  of  the  three.     (Prin.  1.) 

1x3x5 j  5  t  v  ' 

"2"  x  3  x  g  —  31T  But  since  the  value  of  the  fractions  is  not  to 

"fxix!— Iro      *>e  changed,  each  numerator  must  be  multiplied 

-fxixl— M     ky   the    same    multiplier    as   its    denominator. 

Hence,  multiplying  the  terms  of  \  by  3  and  5, 

the  result  is  ^  ;  of  f ,  by  2  and  5,  the  result  is  f  £  ;  and  of  §  by  2 

and  3,  the  result  is  ^#.    Or, 

To  find  the  numerators,  take  such  part  of  the  common  denomina- 
tor 30,  as  the  given  fraction  is  part  of  1.     Thus,  -|  of  30  is  15,  etc. 

Reduce  to  fractions  having  a  common  denominator 


2.  §  andf.    I     4.   f,  |,  andj. 

3.  A  and  f.         5.  ^  f,  and  J. 


6.  3%,  f,  and  \. 

7.  £,  #,  and  ^ 


OPERATION. 

7 

)3 

7 

14 

1 
t 

=  H 

3 

1 

2 

=  » 

2 

x3 

x7  = 

=  42 

A 

=  tt 

8.  Change  |,  f,  and  -^  to  equivalent  fractions  having 
the  least  common  denominator. 

Analysis.— First  find  the  least 
common  multiple  of  the  given  de- 
nominators, which  is  42.  This  must 
be  the  least  common  denominator 
of  the  given  fractions.    (Prin.  2.) 

9.  Change  |,  T\,  and  fj  to  equivalent  fractions  having 
the  least  common  denominator. 

Rule. — 1.  To  reduce  two  or  more  fractions  to  equiva^ 
lent  fractions  having  a  common  denominator. 

Multiply  the  terms  of  each  fraction  by  the  denominators 
of  all  the  other  fractions. 


BEDUCTIOtf.  109 

2.  To  reduce  them  to  their  least  common  denominator. 

I.  Find  the  least  common  multiple  of  the  denominators 
of  the  given  fractions  for  their  least  common  denominator. 

II.  Divide  this  common  denominator  by  the  denomina- 
tor of  each  of  the  given  fractions,  and  multiply  its  numer- 
ator by  the  quotient.     The  products  are  the  new  numerators. 

Mixed  numbers  must  first  be  reduced  to  improper  fractions. 

Eeduce  to  fractions  haying  the  least  common  denomi- 
nator. 


10.  1,  H>  and  £J. 

H-  A>  ih  and  A- 

1&  «,  A,  and  |i. 

13.  m>  i>  U>  and  *J. 


14.  f ,    2f ,    |,  and  1TV 

15.  6J,  A    7,  and  1£. 

16.  H>  Afc  tt,  and  2|. 

17.  W,  AV,  tt,  and  £, 


ADDITION. 

Oli^LL      EXERCISES. 

219.  1.  What  is  the  sum  of  f  and  -§  ?    Of  $  and  J  ? 

2.  How  many  times  1  is  the  sum  of  f ,  -f,  and  f  ? 

3.  Sold  fa  of  an  acre  of  land  to  one  man,  fa  to  another, 
and  fa  to  a  third.     How  much  was  sold  to  all  ? 

How  are  fractions  added  that  have  a  common  denominator  ? 

4.  Mary  paid  $  ■§■  for  some  ribbon,  and  $-§•  for  a  pair  of 
gloves.    How  much  did  she  pay  for  both  ? 

Analysis. — She  paid  the  sum  of  $f  and  $f .    £  is  equal  to  -^,  and 
f  is  equal  to  {$  ;  T^  and  |f  are  |f,  or  1TV    Hence  she  paid  $1^. 

5.  A  man  having  f  of  a  ton  of  coal,  bought  f  of  a  ton 
more.     How  much  had  he  then  ? 

How  are  fractions  added  that  have  different  denominators? 


110  FRACTIONS. 

6.  Henry  gave  $f  for  a  book,  $J  for  a  slate,  and  $£  for 
a  bottle  of  ink.     What  did  he  pay  for  all  ? 

7.  What  is  the  sum  of  J,  J,  and  £  ?    Of  f ,  J-,  and  i  ? 

8.  Find  the  sum  of  £,  £,  and  fa.     Of  £,  f ,  and  fa, 

9.  Find  the  sum  of  -fa,  -§,  and  J.     Of  £,  £,  and  -fa. 
101  A  farmer  sold  3£  tons  of  hay  to  one  man,  and  5| 

to  another.    How  much  did  he  sell  to  both  ? 

Analysis. — The  sum  of  3£  tons  and  5}  tons.     5  and  3  are  8  ;  and 
|  and  f  are  f ,  which  added  to  8  makes  8|  tons. 

11.  A  man  bo  tight  5  J  cords  of  wood  at  one  time,  and 
7  fa  at  another.     How  much  did  he  buy  in  all  ? 

How  are  mixed,  numbers  added  ? 

12.  A  man  paid  125-J  for  a  watch,  and  sold  it  for  $6{ 
more  than  he  gave  for  it.     What  did  he  sell  it  for  ? 

Find  the  sum 


13.  Of   |  and  3f 

14.  Of  5£  and  -}. 

15.  Of  If  and  f . 


16.  Of    2J  and  6|. 

17.  Of    84-  and  fa. 

18.  Of  15J  and  f 


19.  Of  2J-   and  If. 

20.  Of5£   andf 

21.  Of  lfa  and  12£. 


220.  Peikciple.  — Fractions  can  be  added  only  when 
they  have  a  common  denominator,  and  when  they  express 
parts  of  lilce  units. 

WRITTEN     EXERCISES. 

221.  1.  Find  the  sum  of  f ,  fa,  and  -fa. 

operation.  Analysis. — Reduce  the  given  frac- 

| _|_ fa _j_ fa  =  31+g54lft      tions  to  equivalent  fractions  having 
24+35+1  6  __  15  _  1 1         the  least  common  denominator,  which 
*        is  GO  (217,  2).    Then  add  their  nu- 
merators, and  write  the  sum,  75,  over  the  common  denominator  60, 
and  |f  =  1 J  is  the  required  result. 


ADDITION. 


Ill 


2.  What  is  the  sum  of  14},  25 J,  and  7|  ? 


14* 

=  14if 

25| 

=»86H 

n 

=    7*i 

46  +  f4  = 

484, 

Find  the  sum 

3. 

Of  ^  ii  and 

A- 

4. 

Of  |,  J,  and  f . 

Analysis. — The  sum  of  the  frac- 
tions is  ||  =  2\,  which  added  to  the 
sum  of  the  integers  46,  gives  48£, 
the  required  sum. 


5.  Of  42,  31  A,  and  9  A- 

6.  Of204^,50H,  and7^. 


Rule. — I.  Reduce  the  given  fractions  to  equivalent  frac- 
tions having  the  least  common  denominator,  and  write  the 
sum  of  the  numerators  over  the  common  denominator. 

II.  Wlien  there  are  mixed  numbers  or  integers,  add  the 
fractions  and  integers  separately,  then  add  the  results. 


8.  }+?*+»+*=? 

9.  i&fc+2*+lH=? 


10.  -f +  6^  +  211+77=  ? 

11.  H  +  «  +  7i  +  60+tt=? 

12.  1244  +  325^+40^-=? 


13.  Bought  3  pieces  of  cloth  containing  105|,  86},  and 
58-f  yards  respectively  ;  how  many  yards  in  all  ? 

14.  If  it  takes  5J  yards  of  cloth  for  a  coat,  3£  yards  for 
a  pair  of  pantaloons,  and  f  of  a  yard  for  a  vest,  how  many 
yards  does  it  take  for  all  ? 

15.  Four  cheeses  weighed  respectively  46|,  48|,  49^, 
and  57J  pounds.     What  was  their  entire  weight  ? 

16.  What  number  is  that  from  which  if  24^-  is  taken, 
the  remainder  is  63f|  ? 

17.  A  farm  is  divided  into  4  fields :  the  first  contains 
29T\-  acres,  the  second  50|-J  acres,  the  third  41f  acres,  and 
the  fourth  69}  acres.     How  many  acres  in  the  farm  ? 


112  FEACTIONS. 

SUBTEACTION. 

ORATj     exercises. 

222.  1.  What  is  the  difference  between  f  and  -f  ? 

2.  What  is  the  difference  between  ^  and  -£%  ? 

How  is  one  fraction  subtracted  from  another,  each  having  the 
same  denominator  ? 

3.  A  gentleman  who  owned  a  sail-boat  sold  -fa  of  it. 
^  What  part  did  he  still  own  ? 

4.  A  boy  having  $£,  gave  %\  for  a  neck-tie.    What  had 

he  left  ? 

Analysis  .— He  had  left  the  difference  between  $|  and  %\.  £  is 
equal  to  ^,  and  i  equals  ^  ;  ^  less  ^  are  ^. 

5.  A  man  owning  f  of  an  acre  of  ground,  sold  %  of  an 

acre.    What  part  remained  ? 

How  is  one  fraction  subtracted  from  another  having  a  different 
denominator  ? 

6.  Subtract  -J-  from  £  ;  f  from  \  ;  f  from  -&■. 

7.  Find  the  difference  between  TV  and  f  ;  §  and  \. 

8.  From  a  piece  of  cloth  containing  12|-  yards,  5 \  yards 

were  cut.     How  many  yards  remained  ? 

Analysis. — The  difference  between  12|  yards  and  5|  yards.  \ 
from  i  leaves  f ,  and  5  from  12  leaves  7.    Hence  7|  yards  remained. 

9.  If  a  ton  of  coal  costs  $7|,  and  a  cord  of  wood  $4  J, 
what  is  the  difference  in  their  cost  ? 

How  is  one  mixed  number  subtracted  from  another  ? 

10.  What  is  the  value  of  3f- 2-fc?    8±-2£?    G|£— |  ? 

223.  Pkinciple. — Fractions  can  be  subtracted  only 
when  they  have  a  common  denominator,  and  when  they 
express  parts  of  like  units. 


SUBTRACTION.  113 

m 

WRITTEN     EXERCISES. 

224.  1.  From  £  subtract  ^. 

operation.  Analysis. — Reduce  the 

7  4-_35  il  =  3  5-1  2  _  23      given  fractions  to  equiva- 

t       T5       4  6-       rr  <FT  rs"     ient  fractions  having  the 

least  common  denominator.     Hence  f  §  —  £§  =  |f . 

2.  From  134J  take  76£. 
operation.  Analysis. — Reduce  ^  and  f  to  equivalent 

1  o  At  1  Q  A  8       fractions  having  the  least  common  denomi- 

*  ~  ^     nator.     As  ^f  cannot  be  taken  from  ¥8T,  take 

7  6  -£  =     76f|      1  or  2  4  from  i34?  leaving  133,  and  add  it  to  &, 

5  7  fj     making  f  f .  Then  $f  from  § f  leaves  |f,  and  76 

from  133,  leaves  57.   Hence  57||  is  the  result. 


3.  From  -fa  take  £$. 

4.  From  -^  take  ■£%. 

5.  From  |f  take  ^V 


6.  From  36-f-  take  1  Of 

7.  From  112^  take  56. 

8.  From  204&  take  39^. 


Eule. — I.  Reduce  the  given  fractions  to  equivalent  frac- 
tions having  the  least  common  denominator,  and  write  the 
difference  of  the  numerators  over  the  common  denominator. 

II.  When  there  are  mixed  numbers,  subtract  the  frac- 
tional and  integral  parts  separately,  and  add  the  results. 

If  the  mixed  numbers  are  small,  they  may  be  reduced  to  improper 
fractions  and  subtracted  according  to  the  usual  method. 

Find  the  difference  between 

9.     |  and  ^.        11.  ft  and  2f        13.  63J  and  71f 
10.   If  and  £  12.  16  and  3^T.      14.  106  and  95^f 

15.  From  -&  take  ^.        17.  From  410^T  take  226f 

16.  From  16T\  take  ^.     18.  From  428|£  take  180fJ. 
19.  A  farmer  having  208  acres  of  land,  sold  92^  acres. 

How  many  acres  had  he  left  'i 


114  FRACTION  S. 

ADDITION    AND    SUBTRACTION. 
ORAL     EXERCISES. 

225.  1.  How  much  less  than  2,  is  £+£  ? 

2.  How  much  greater  than  2,  is  § +  £  +  l£? 

3.  What  is  the  difference  between  4  and  2|  ?  5^  and  7-f-  ? 

4.  Mr.  Smith  sold  J  of  his  farm  to  one  man,  J  to 
another,  and  -J  to  a  third.     What  part  had  he  left  ? 

5.  Paid  $6 -J-  for  a  ton  of  coal,  and  $3  J-  for  a  load  of  wood. 
What  change  must  be  returned  for  a  ten-dollar  bill  ? 

6.  What  is  the  difference  between  If  -f  f  and  5^  ? 

7.  What  is  the  difference  between  A  +  i  and  i+i? 

Find  the  second  member  of  the  following  equations  : 


8.  |+W=? 

10.  3-(*-i)=? 

11.  6i  +  t-lf=? 

12.  12-(8-2|)  +  H=? 


13.  (26-14)-(*  +  3})=? 

14.  (l!  +  2i)-(£  +  f)=? 

15.  (5-34)  +  (9A-6)=? 

16.  8A-2m  +  10=? 

17.  12^  +  9i-li-2i=? 


WRITTEN     EXAMPLES. 

226.  1.  The  sum  of  two  numbers  is  124J-,  and  the  less 
is  36^.     What  is  the  greater  ? 

2.  What  number  added  to  147^  will  make  21 6£? 

3.  What  number  added  to  307++ 210J  will  make  700f  ? 

4.  What  number  must  be  added  to  the  difference  of 
186 f-  and  214f  to  make  1042£f  ? 

5.  What  fraction  added  to  the  sum  of  •£,  ^,  and  ^, 
will  make  |f|  ? 

6.  What  must  be  added  to  $ ,  that  the  sum  may  be  J-J  ? 


MULTIPLICATION,  115 

7.  Bought  a  quantity  of  barrel  staves  for  $160-§,  and  of 
lumber  for  $1136f .  Sold  the  staves  for  $205£  and  the 
lumber  for  $1240^.     What  was  the  whole  gain  ? 

8.  A  man  bought  a  ton  of  hay  for  $15|,  a  barrel  of  flour 
for  $9^,  and  a  barrel  of  apples  for  $3^.  What  change 
should  be  given  to  him  for  3  ten-dollar  bills  ? 

Complete  the  following  equations  : 

10.  8§  +  2%-5&=? 

11.  48-(16*— 3*)=? 


12.  41j  +  56-24A-4H=  ? 

13.  120— 51f  +  90i$— f=? 

14.  342-(2TA  +  i-9)=? 

15.  176A-  +    132£  -  26f£  -  1U  =  ? 

16.  $1000  -  $500  +  non-Ms  +  $91-rV  =  ? 

MULTIPLICATION. 

,  22*7«  When  one  factor  is  a  fractional  number. 

ORAL      EXERCISES. 

1.  What  part  of  a  mile  is  3  times  J  of  a  mile  ? 

2.  What  part  of  a  dollar  is  4  times  -J-  of  a  dollar  ? 

3.  How  many  times  1  is  3  times  f  ?    4  times  }  ? 

4.  At  f-f  a  pound,  what  will  4  pounds  of  tea  cost  ? 

Analysis.— Four  pounds  will  cost  4  times  $jj,  or  $28°,  equal  to 
$2*-. 

5.  At  l-J  a  bushel,  what  is  the  cost'of  6  bushels  of  oats  ? 
Of  7  bushels  ?     Of  8  bushels  ?     Of  9  bushels  ? 

6.  If  a  horse  eat  ^  of  a  bushel  of  grain  in  a  day,  how 
much  will  4  horses  eat ?   6  horses?   8  horses?   10  horses? 

7.  What  cost  12  baskets  of  pears,  at  If  a  basket? 
With  each  class  of  oral  questions  in  Art.  227,  the  pupil  may 

solve  the  corresponding  written  examples  on  pages  118  and  119. 


116  FRACTIONS. 

8.  What  cost  9  pounds  of  butter  at  $-£■  a  pound  ? 

9.  Show  that  multiplying  the  numerator  of  ¥3F  by  4 
multiplies  the  fraction  by  4. 

10.  Show  that  dividing  the  denominator  of  ^  by  4 
multiplies  the  fraction  by  4. 

How  many  ways  to  multiply  a  fraction  by  an  integer  ? 

11.  Multiply  A  by  5  ;   ^  by  6  ;    *  by  5  ;   A  by  8. 

12.  At  $4f  a  box,  what  will  5  boxes  of  raisins  cost  ? 

Analysis. — They  will  cost  5  times  $4f .    5  times  $|  are  $3f,  and 
5  times  $4  are  $20.     $20  +  $3|  =  $23f.     Hence,  etc. 

13.  At  7£  cents  a  pound,  what  will  9  pounds  of  rice  cost  ? 

14.  At  $9-£  a  barrel,  what  is  the  cost  of  6  barrels  of 
flour  ?     Of  8  barrels  ?     Of  9  barrels  ?     Of  10  barrels  ? 

15.  What  will  8  yards  of  cloth  cost,  at  $5T\  a  yard  ? 

16.  Multiply  7|  by  9  ;   9^  by  6  ;   10#  by  7  ;   12^  by  8. 

17.  What  is  £  of  12  yards  ?     i  of  24  men  ?     %  of  $30  ? 

18.  Multiplying  by  J,  -J-,  J,  ^,  etc.,  is  the  same  as 

dividing  by  what  integers  ? 

When  a  fractional  part  of  an  integer,  or  of  a  fraction,  is  to  be 
taken,  the  word  of,  and  not  times,  should  be  used. 

19.  At  $7  a  ton,  what  will  f  of  a  ton  of  coal  cost  ? 

Analysis.— It  will  cost  f  of  $7,  or  3  times  |  of  $7.     |  of  $7  is 
$1|,  and  3  times  $1|  are  $5^.     Hence,  etc. 

20.  At  $12  a  gross,  what  will  -J  of  a  gross  of  butts  cost  ? 

21.  What  will  -fo  of  a  ton  of  hay  cost,  at  $15  a  ton  ? 

22.  What  is  4  of  6?     £  of  2  ?    f  of  8  ?    A  of  9  ? 

23.  At  $5  a  yard,  what  will  $  of  a  yard  of  cloth  cost  ? 

24.  If  a  man  can  build  a  wall  in  28  days,  in  what  time 
can  he  build  $  of  it  ?    f  of  it  ?    £  of  it  ? 

25.  If  an  acre  of  land  produce  45  bushels  of  corn,  how 
much  will  f  of  an  acre  produce  ?    -£  ?    f  ?    %?    ■£$? 


MULTIPLICATION. 


117 


26.  What  is  f  of  $5?     Of  $7  ?     Of  $16  ?    Of  $25  ? 
.    27.  Multiply  50  by  }  ;    49  by  f  ;    63  by  &  ;   81  by  -&. 

28.  In  $1  are  100  cents ;  how  many  cents  in  £  of  a 
dollar?    InJ?    *!    f?    &?    -&?    ^? 

29.  How  many  cents  in  -fw  of  a  dollar  ?   In  ^-  ?  f  ?  £  ? 

30.  Which  is  greater,  $  of  15,  or  15  x  i  ? 

31.  Show  that  a  fraction  of  an  integer  equals  the  pro- 
duct of  the  integer  by  the  fraction. 

How  is  an  integer  multiplied  by  a  fraction  ? 

32.  At  $12  a  ton,  what  will  5f  tons  of  cheese  cost  ? 

Analysis.— It  will  cost  5|  times  $12.    5  times  $12  are  $60,  and 
|  of  $12  are  $4|,  which  added  to  $60,  make  $64|. 

33.  At  15  cents  each,  what  will  4f  melons  cost  ? 

34.  What  will  7f  weeks'  board  cost,  at  $9  a  week  ? 

35.  How  much  is  6f  times  12  ?    5f  times  20  ? 

36.  Multiply  4  by  8£  ;    6  by  7| ;    8  by  9^. 


What 

37.  Is  TV  of  10  gallons  ? 

38.  Is  |  of  47  pounds  ? 

39.  Is  f  of  90  rods? 

40.  Is  J  of  56  days  ? 
Find  the  value 

45.  Of  §  x  15. 

46.  Of  28  xf 

47.  Of  56  xf 

48.  Of  ^x7. 


How  many 

41.  Are  10  times  6f  tons  ? 

42.  Are  9  times  11-&  miles? 

43.  Are  7f  times  12  men  ? 

44.  Are  12^  times  9  minutes? 


53.  Of^+f  x5. 

54.  Offt-2xf 

55.  Of  4^+2  x5f. 

56.  Of  9x9|-f20J, 


49.  Of  fx7  +  £. 

50.  Of  1x9— 2f 

51.  Of  27x|  +  3i. 

52.  Of  6f  x7— If 

228.  Principles. — 1.  Multiplying  the  numerator  or 
dividing  the  denominator  multiplies  the  fraction.  (200, 1.) 

2.   The  product  of  an  integer  by  a  fraction  is  equal  to 
such  part  of  the  integer  as  the  fraction  is  of  a  unit. 


118 


FRACTIONS: 


WRITTEN    EXERCISES, 


229.  1.  Multiply  fr  by  9. 


OPERATION. 


*x9=W=tf=H 
Or, 


Analysis.— Multiply  the 
numerator  7  by  9,  or  divide 
the  denominator  27  by  9  ; 
either  operation  will  give 
2^,  the  required  result. 
(Pkin.  1.) 

By  using  the  vertical 
line  and  cancellation,  both  operations  are  combined  and  shortened. 
In  the  first  operation,  the  number  of  parts  or  of  fractional  units 
is  increased,  while  their  size  or  value  remains  the  same  :  in  the 
second  operation,  the  size  of  the  parts  is  increased,  while  their 
number  remains  unchanged. 


Or, 

zn 

7 

0 

3 

7 

n 

In  like  manner  multiply 


2.     A  by  12. 

5.     ^  by  15 

3.     fj  by  9. 

6.    f&  by  36 

4.     tf  by  13. 

V.    T*Aby21 

8. 

9. 

10. 


tU  by  17. 
Hi  by  22. 
tt   by  44. 


11.  Multiply  72  by  |. 


OPERATION. 


72  x  £=72-^9x4=32 
Or,  72x£=-*F±=32 


Or, 

n 


Analysis. — To  multi- 
ply 72  by  |,  is  to  find  £ 
of  72.  |  of  72  is  4  times 
i  of  72,   which    is    32. 


32       (Prin-  2.) 


Find  the  product 


12.  Of  75  by  A- 

13.  Of  7by^r. 

14.  Of  56  by  ^. 


15.  Of  168  by  £f- 

16.  Of  200  by  ^. 

17.  Of  315  by  if 


18.  Of  19  by  Jf 

19.  Of  448  by  ^. 

20.  Of  572  by  ■&. 


A  fraction  is  multiplied  by  a  number  equal  to  its  denominator  by 
cancelling  the  denominator.    Thus,  |x8  =  7. 

Cancelling  a  factor  of  the  denominator  multiplies  the  fraction  by 
that  factor.    Thus,  -&  x  4  =  £ . 


MULTIPLICATION. 


119 


21.  Multiply  17f  by  6. 


OPERATION. 


17f 
6 


Or,  17f  =  -H^ 
155 


103* 


310 


1031 


Multiply 

22.  1271  by  12. 

23.  851  by  15- 


Analysis.— To  multiply  17f  by  6, 
multiply  the  fraction  § ,  and  the  in- 
teger 17  separately  and  add  their 
products,  which  gives  103  £,  the  re- 
quired product.     Or, 

Reduce  the  mixed  number  to  an 
improper  fraction,  and  multiply  as  in 
Ex.  1,  which  gives  the  same  result. 


24.  128^  by  42. 

25.  2461    ty  16. 


26.  314^  by  48. 

27.  750^  by  17. 


28.  Multiply  140  by  9f 


140 

9f 

931 
1260 
13531 


operation.  Analysis.— To  multiply  140  by  9|, 

multiply  by  the  fraction  f ,  and  by  the 
integer  9,  separately,  and  add  their 
products,  which  gives  1353£,  the  re- 
quired product.    Or, 

Reduce  the  mixed  number  to  an 
improper  fraction,  and  multiply  as  in 
1  o  O  6 1     ex>  i}  which  gives  the  same  result. 


Or,  9f  =  ¥ 
140 
29 


4060 


Multiply 

29.  96  by  12f . 

30.  216  by  16$. 


31.  304  by  24^.      33.  560  by  23^- 

32.  198  by  18f        34.  715  by  14^. 

35.  Multiply  327^  by  ™  J  2466  by  84i  5  ?59  by  M- 

36.  What  will  120  dozen  of  hose  cost  at  |4|  a  dozen? 

37.  At  $20  a  ton,  what  will  H  of  a  ton  of  hay  cost  ? 

38.  If  a  city  lot  is  worth  $3145,  what  is  ^  of  it  worth  ? 

39.  What  will  142  yards  of  curbing  cost  at  |6f  a  yard  ? 

40.  At  $|f  a  yard,  what  is  the  cost  of  8  yards  of  cloth  ? 
Of  24  yards  ?    Of  64  yards  ?    Of  120  yards  ? 


120  FRACTIONS. 

230.  When  both  factors  are  fractional  numbers. 


OliAZ      EXERCISES 


1.  A  boy  having  \  of  a  melon,  gave  \  of  it  to  his  sister. 
What  part  of  the  melon  did  she  receive  ? 

2.  Whatpartof  1  is  £  of  |?     Is£of£?     Is  J  of  |? 

3.  What  part  of  lis  \  of  f?    |of£?    Joff?    ioffr 

4.  Which  is  greater,  \  of  \,  or  -J  of  J  ?    J  of  £,  or  J  of  f  ? 
±ofi,  or£ofi? 

5.  If  I  own  -f  of  an  acre  of  land,  and  sell  \  of  it,  what 
part  of  an  acre  do  I  sell  ?    What  part  do  I  retain  ? 

6.  If  a  yard  of  silk  is  worth  $-§,  what  is  \  of  a  yard  worth  ? 

7.  A  boy  having  $j  gave  |-  of  it  for  a  knife.    What  part 
of  a  dollar  did  he  pay  for  the  knife  ? 

Analysis.— He  paid  f  of  $|,  or  5  times  \  of  %\.  \  of  $|  is  $^, 
and  5  times  %^  are  Jf,  or  $f. 

8.  At  If  a  gallon,  what  will  J  of  a  gallon  of  syrup  cost  ? 

Fractions  with  the  word  of  between  them  are  sometimes  called 
Compound  Fractions.  The  word  of  is  equivalent  to  the  sign  (  x  )  of 
multiplication.     Thus,  f  of  f  =  f  x  f  ;  £ o/  9  =  f  x  9,  etc. 

9.  Whatisfoff?    foff?    foff?    foff? 

10.  Whatis^xf?    -&xf?    fxA?    IxA? 

11.  A  man  owning  £  of  a  mill,  sold  £  of  his  share  to 
his  brother.    What  part  of  the  mill  did  each  then  own  ? 

12.  At  $8|  a  barrel,  what  will  f  of  a  barrel  of  flour  cost  ? 

Analysis.— It  will  cost  £  of  $8f,  or  3  times  \  of  $8|.    \  of  $8|  is 
$21-,  and  3  times  $2£  are  $6|. 
Or,  $8|  equal  $^,  and  f  of  $\*  =  $sp  =  $6|. 

13.  At  $9£  a  case,  what  will  f  of  a  case  of  slates  cost  ? 

14.  At  $5 \  a  yard,  what  will  \  of  a  yard  of  cloth  cost  P 


MULTIPLICATION, 


121 


15.  How  much  is  |  of  7£  miles  ?    f  of  24£  pounds  ? 

16.  If  a  man  travel  26|  miles  in  a  day,  how  far  does  he 
travel  in  £  of  a  day  ?    In  |  ?     Inf?    Inf? 

17.  At  $|  a  yard,  what  will  6 J-  yards  of  flannel  cost? 

Analysis. — It  will  cost  6^  times  $|.    6  times  $|  are  $4,  and  £  of 
$|  is  $-£,  which  added  to  $4,  makes  $4£. 
Or,  6£  =  \5-,  and  \*  of  $|  =  $2/  =  $4|. 

18.  At  lyTg-  a  pound,  what  will  8J  pounds  of  tea  cost  ? 

19.  What  will  9|  pounds  of  beef  cost,  at  $}-  a  pound  ? 

20.  If  a  man  hoe  -J  of  an  acre  of  corn  in  a  day,  how  many 
acres  can  he  hoe  in  5£  days  ?     In  6f  days  ?     In  8£  days  ? 

21.  What  will  2£  yards  of  silk  cost,  at  $3£  a  yard  ? 

Analysis.— 2J=|,  and  3^=^  '>  and  ¥  *  t=¥=$7i 
Reduce  mixed  numbers  to  improper  fractions,  and  then  proceed 
as  in  multiplying  one  fraction  by  another. 

22.  What  cost  5f  yards  of  alpaca,  at  $lf  a  yard  ? 

23.  Multiply  2£  by  3*  ;  7£  by  If ;  5J  by  2|. 

Find  the  value 


24.  Of  f  of  |  of  a  mile. 

25.  Of  }  of  |  of  J  of  $1. 

26.  Of  |  of  4^-  leagues. 

27.  Of  j-  of  6f  pecks. 
Find  the  result  of 


28.  Of  7J  times  *§, 

29.  Of  9|  times  f  of  a  rod. 

30.  Of  2 J  times  5£  acres. 

31.  Of  Q-  times  |  of  2J  feet 


32. 

4xf. 

36. 

10-f  of  3J. 

40. 

2Jxf~|oflf. 

33. 

lfx|. 

37. 

6J  +  2Jx|. 

41. 

16-^x121. 

34. 

xVx6f 

38. 

H-iof2}. 

42. 

4i  +  fofH-^. 

35. 

Hxlf 

39. 

»x*  +  8f 

43. 

if  of  V-+2*X0. 

231.  Prist. — 7%e  product  of  a  fraction  by  a  fraction  is 
such  apart  of  either  factor  as  the  other  is  of  a  unit. 


122 


FRACTIONS 


WRITTEN     EXERCISES, 


232.  1.  Multiply  £  by  £ 


OPERATION. 


Ax* 


JA=A 

Or,  Ax  1= A 


Analysis.— To  multiply  -^  by  £  is  to  find 
£  of  ^,  which  is  7  times  £  of  ^-,  and  £  is 
found  by  multiplying  the  denominator  (200, 
1.)  Hence  £  of  TB¥  is  1^,  and  £  of  T5¥  is  7 
times  jfj,  or  -fifo  =  T5^,  the  required  product. 


In  like  manner  multiply 


2.  ttbyff- 
3-  «by«. 


4.  AbyH- 

5.  Hbyfi- 


6-  At  ty  7*>- 

7-  A  by«f 


8.  Find  the  product  of  ^  ^,  5|,  and  2. 


OPERATION. 


H  x 


** 


n  x  3 

3 


2_8 
X  1""9 


Or, 

J0 

% 

3M 

4 

3 

10 

2 

9 

8  =  | 

Analysis.  — Change 
the  mixed  number  §\ 
to  an  improper  frac- 
tion, and  the  integer 
2  to  the  form  of  a 
fraction,  then  multi- 
ply as  in  Ex.  1. 


Find  the  value  of  the  following  expressions  : 


9.  ^0fl2Jxiof7i. 

10.  f  of  96  x  A  of  26f. 

11.  J  off  of  21£x2f 

12.  42£  x  5£  times  6f . 


13.  $of36£xf  of  9. 

14.  fof  91x72f 

15.  ^  of  63J  by  \  of  |f. 
1G.  Aof  21xA°fJf  °f£ 


From  the  preceding  principles  and  operations  is  derived 
the  following  general 

Rule. — I.  Reduce  all  integers  and  mixed  numbers  to 
improper  fractions. 

II.  Multiply  together  the  numerator  sy  for  the  numera- 
tor ;  and  the  denominators,  for  the  denominator  of  the 
product.     Or, 


MULTIPLICATION.  123 

I.  Multiply  by  the  numerator  of  the  fractional  multi- 
plier and  divide  by  the  denominator. 

II.  When  the  multiplier  is  a  mixed  number,  multiply  by 
the  fractional  and  integral  parts  separately  and  add  their 
products. 

Cancel  all  factors  common  to  numerators  and  denominators  before 
multiplying,  thus  shortening  the  operation,  and  obtaining  the 
answer  in  its  lowest  terms. 

Find  the  cost 

17.  Of  15  cords  of  bark,  at  $4§  a  cord. 

18.  Of  24J  pounds  of  tea,  at  $-J  a  pound. 

19.  Of  80  yards  of  cloth,  at  $±ft  a  yard. 

20.  Of  21f  bushels  of  corn,  at  $ft  a  bushel. 

21.  Of  |  of  5|  tons  of  hay,  at  $15^  a  ton. 

22.  Of  18|  barrels  of  crude  oil,  at  $7$  a  barrel. 

23.  Of  i  of  18 1  yards  of  silk,  at  J  of  $5  a  yard. 

24.  Of  126  pounds  of  beef,  at  9J-  cents  a  pound. 

25.  Of  36^  tons  of  railroad  iron,  at  $62J  a  ton. 

26.  Of  35  horses,  at  $205£  each. 

27.  Of  £  of  156 1  acres  of  land,  at  £  of  $54^  an  acre. 

28.  Of  28f  bushels  of  sweet  potatoes,  at  $lf  a  bushel. 

29.  Of  |  of  a  yard  of  satin,  at  IJ-J  a  yard. 

30.  Of  ft  of  an  acre  of  land,  at  $125  an  acre. 

31.  Of  7^9o  tons  of  middlings,  at  $26f  a  ton  ? 

32.  Paid  $365-J  for  a  horse,  and  sold  him  for  f  of  what 
he  cost.     What  was  the  loss  ? 

33.  When  peaches  are  worth  $}  a  basket,  what  are  126  j 
baskets  worth  ? 


34.  Find  the  value  of  (129  —  76f )  x  ft  of  12}  —  2f + 
21J-  x  6f 


124  FRACTIONS. 

DIVISION. 

233.  When  the  divisor  is  an  integral  number. 
OJtA.1,     exercises. 

1.  If  |  of  an  acre  of  land  is  divided  into  3  equal  lots, 
what  part  of  an  acre  does  each  lot  contain  ? 

2.  i  of  £  is  what  part  of  1  ?    |  of  £  ?    J- of-}?    iof^? 

3.  Dividing  by  3,  4,  and  5  is  the  same  as  multiplying 
by  what  fractions  ? 

4.  How  much  is  ^xi?    ft+B?    AxJ?    -&^4  ? 

5.  If  4  slates  cost  $|,  what  will  1  slate  cost  ? 

6.  If  5  boxes  of  figs  cost  $£,  what  will  1  box  cost  ? 

7.  Divide  -J  of  a  barrel  of  flour  equally  among  3  families. 
What  part  of  a  barrel  will  each  family  receive  ? 

8.  What  is  i  of  -ft.  ?    -ft  divided  by  8  ? 

9.  What  is  the  quotient  of  Jf  divided  by  2  ?    by  3  ? 

10.  Show  that  dividing  the  numerator  of  ^  by  3  divides 
the  fraction  by  3. 

11.  Show  that  multiplying  the  denominator  of  f  by  3 
divides  the  fraction  by  3. 

How  many  ways  to  divide  a  fraction  by  an  integer? 

12.  DivideAt)y4;Mby5;  -ft  by  7  ;  *  by  9. 

13.  If  6  pounds  of  sugar  cost  If,  what  will  1  pound  cost  ? 

14.  Divide  |  by  f  of  20  ;  ff  by  f  of  15  ;  $  by  i  of  40. 

15.  At  $4  a  yard,  how  many  yards  of  silk  can  be  bought 
for  $21$? 

Analysis. — As  many  yards  us  $4  is  contained  times  in  $21f,  or  J 
of  21  f  =  I  of  if*,  or  $A  =  5f  times. 

Or,  4  is  contained  in  21  f ,  5  times  and  If  or  -^  remainder ;  \  of  -^ 
is  f ,  which  added  to  5  makes  5^.    Hence,  etc. 


DIVISION 


125 


16.  If  a  man  walks  18|  miles  in  4  hours,  how  far  does 
he  walk  in  1  hour  ?    In  3  hours  ?     In  5  hours  ? 

17.  How  many  times  will  16$  gallons  of  cider  fill  a  ves- 
sel that  holds  3  gallons  ? 

18.  What  cost  1  pound  of  sugar,  if  6  pounds  cost  $1|  ? 

19.  If  a  hoy  earn  $12£  in  10  days,  how  much  does  he 
earn  in  1  day  ?    In  4  days  ?     In  5  days  ?    In  7  days  ? 

20.  Divide  8J-  by  5  ;    lOf  by  7  ;    18J  by  12  ;  4J  by  8. 

21.  Divide  J  of  21,  by  -f  of  10  ;    I  of  29,  by  |  of  63. 

Find  the  value  of 

5. 


22. 

tt- 

-5. 

23. 

*-* 

-8. 

24. 

Itt- 

4-& 

25. 

*H 

-7. 

26.  9f 

27.  13|-j-4-2f. 

28.  27}-=- 6  + If. 

29.  ft-5-7xlO. 


30.  1^-9  xf 

31.  5|x4  +  13i-j-3. 

32.  tt-5-7Xfof  lj. 

33.  A-ixi-^4. 


234.  Principle. — Dividing  the  numerator  or  multi- 
plying the  denominator  divides  the  fraction.     (200,  2.) 

Always  divide  the  numerator  when  it  is  a  multiple  of  the  divi- 
sor ;  otherwise  multiply  the  denominator. 


WRITTEN    EXERCISES, 


235.  1.  Divide  £f  by  6. 


OPERATION. 


Or,  f|-6=^5T=A 


Analysis.— First,  to  divide  if  by 
6,  divide  the  numerator  of  the  frac- 
tion by  6,  which  gives  -fa  for  the  quo- 
tient.   Or, 

Multiply  the  denominator  of  the 
fraction  by  6,  which  gives  the  same  result. 

In  the  first  operation,  the  number  of  parts  or  of  fractional  units 
is  diminished,  while  their  size  or  value  remains  the  same  ;  in  the 
second  operation,  the  number  of  the  parts  remains  unchanged, 
while  their  size  is  diminished. 


120 


FRACTIONS 


Divide 

2.  if  by  8. 

3.  Ml  by  25. 


H  by  18. 

m  by  21. 


iW*rby64. 
1%  by  35. 


50 


8.  Divide  50f  by  8. 

OPERATION. 

Or,    8 


50f 


6.A 


Analysis. — Reduce  the  mixed 
number  to  an  improper  fraction 
and  divide  as  in  Ex.  1.     Or, 

Divide  as  in  simple  numbers  ; 
8  is  contained  in  50f ,  6  times 
and  a  remainder    of    2| ;    2% 


equal  -1/,  which  divided  by  8  gives  £§  or  y^,  which  added  to  the 
partial  quotient  6  gives  6^j,  the  required  quotient. 

What  is  the  quotient 

9.  Of  4BJ  divided  by  7  ?  12.  Of  248£  divided  by  48  ? 

10.  Of  128H-  divided  by  8  ?  13.  Of  306^  divided  by  25  ? 

11.  Of  85^  divided  by  21  ?  14.  Of  510|  divided  by  30  ? 

15.  If  20  pounds  of  rice  cost  $lf,  what  is  the  cost  of 
1  pound  ? 

16.  The  product  of  two  numbers  is  72$,  and  one  of 
them  is  14  ;  what  is  the  other  ? 

17.  If  ff  of  an  acre  produce  25  bushels  of  wheat,  what 
part  of  an  acre  will  produce  1  bushel  ? 

18.  If  12  ploughs  cost  $124},  what  is  the  cost  of  each  ? 

19.  What  number  multiplied  by  48  produces  694f  ? 

20.  If  54  horses  cost  $4622f,  what  is  the  cost  of  each  ? 

21.  The  product  of  two  numbers  is  1248|,  and  one  of 
the  numbers  is  32  ;  what  is  the  other  ? 

22.  If  11  men  consume  }  of  lOOf  pounds  of  meat  in 
1  week,  how  much  does  1  man  consume  in  the  same  time  ? 

23.  What  is  the  weight  of  4  tubs  of  lard,  if  12  tubs 
weigh  528J  pounds  ? 


division.  127 

236,  When  the  divisor  is  a  fractional  number. 

ORA.E    EXERCISES. 

1.  How  many  halves  in  1  pound  ?    In  4  pounds  ? 

2.  lis  how  many  times  J?    i?    J?    |?    -J?    |? 

3.  What  is  the  quotient  of  1  divided  by  £  ?  £  ?  J  ?  }  ? 

4.  At  $•$■  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  $6  ? 

Analysis. — As  many  yards  as  $f  is  contained  times  in  $6.  6  is 
equal  to  -3/,  and  4  fifths  is  contained  in  30  fifths  7^  times. 

Or,  $|  is  contained  in  $1,  £  times,  and  in  $6,  6  times  £  or  &?-, 
equal  to  7^  times.     Hence,  etc. 

5.  If  a  boy  earn  $f  a  day,  in  what  time  will  he  earn  $5  ? 

6.  At  $f-  a  pound,  how  much  tea  can  be  bought  for  $9  ? 

7.  1  is  how  many  times  f  ?   j?!?^?!?^.?^? 

8.  3  are  how  many  times  |  ?    f  ?    f  ?    J  ?    ^  ?    ^  ? 

9.  If  a  horse  eat  f  of  a  bushel  of  oats  in  a  day,  in  how 
many  days  will  he  eat  6  bushels  ?  8  bushels  ?  9  bushels  ? 

10.  Divide  12  by  J  ;     16  by  f  ;    25by£;     14  by  f 

11.  If  |  of  a  bale  of  hay  cost  $12,  what  will  1  bale  cost  ? 

Analysis. — Since  §  of  a  bale  cost  $12,  £  of  a  bale  will  cost  £  of 
$12,  or  p£,  and  1  bale,  or  f ,  will  cost  6  times  $*<?-,  or  $7g2-,  equal  to 
$14|. 

12.  What  will  1  ton  of  coal  cost,  if  f  of  a  ton  cost  $7  ? 

13.  How  many  times  is  f  contained  in  6  ?  In  8  ?  In  11? 

14.  How  many  times  is  f  contained  in  f  of  16  ? 

15.  How  many  turkeys  can  be  bought  for  $9  at  $1|  each? 

16.  How  many  garments  can  be  made  from  15  yards 
of  cloth,  if  each  garment  contains  2£  yards  ? 

How  is  an  integer  divided  by  a  fraction  ? 


128  FRACTIONS. 

17.  How  many  times  is  1£  contained  in  5?    2f,  in  9? 

18.  How  many  times  is  2  eighths  of  a  yard  contained 
In  6  eighths  of  a  yard  ?    f  of  a  mile,  in  f  of  a  mile  ? 

19.  At  $T3¥  each,  how  many  pine-apples  can  be  bought 

for$A?     For$A?    For$H?    For  $41? 

20.  How  many  times  %  in  -f  ?    -^  in  |£  ?    -^  in  ^  ? 

How  is  a  fraction  divided  by  a  fraction  when  they  have  a  com- 
mon denominator  ? 

21.  At  $f  a  pound,  how  much  tea  can  be  bought  for  %\  ? 

Analysis. — As  many  pounds  as  $|  is  contained  times  in  $f .  $f 
equals  $^,  and  %\  equals  %\% ;  8  twentieths  is  contained  in  15 
twentieths  1|  times. 

Or,  $|  is  contained  in  $1,  |  times,  and  in  £  of  $1,  f  of  f  or  ^, 
equal  to  If  times.     Hence,  etc. 

22.  How  many  pounds  of  honey  at  %\  a  pound,  can  be 
bought  for  *|  ?    For  If?    For  If?    Forl^? 

23.  Divideibyf  ;  J  by*;  \  by  f  ;  %  by  \  ;  ^  by  J. 

How  is  a  fraction  divided  by  a  fraction  when  they  have  not  a 
common  denominator? 

24.  In  2 \  acres  of  land,  how  many  building  lots  of  f  of 

an  acre  each  ? 

Reduce  mixed  numbers  to  improper  fractions,  then  divide  as  you 
divide  one  fraction  by  another. 

25.  At  %^  a  yard,  how  many  yards  of  cambric  can  be 
bought  for  $J?     For$|?    For  *J-  ?    For$3J? 

26.  At  If  a  bushel,  how  many  bushels  of  onions  can  be 
bought  for  $4£  ?     For  $5 J?    For$3f? 

27.  If  a  man  chop  \\  cords  of  wood  in  a  day,  in  how 
many  days  can  he  chop  5 \  cords  ?   10£  cords  ?    12  cords  ? 

28.  How  many  times  will  4|  gallons  of  kerosene  fill  a 
can  that  holds  \  of  |  of  1  gallon  ? 


DIVISION. 


129 


29.  If  |  of  a  box  of  figs  cost  I},  what  will  1  box  cost  ? 

Analysis. — Since  2  fifths  of  a  box  cost  $|,  1  fifth  of  a  box  will 
cost  |  of  $|  or  $rV,  and  1  box  or  f  will  cost  5  times  $T7^  or  $ff 
equal  to  $2^-. 

30.  If  f  of  a  yard  of  cloth  cost  $f ,  what  will  1  yard  cost  ? 

31.  What  cost  1  quart  of  wine,  if  -^  of  a  quart  cost  l-^f-  ? 


What  is  the  quotient 

32.  Of  f  divided  by  6  ? 

33.  Of  2|  divided  by  7  ? 

34.  Of  16f  divided  by  10? 


How  many  times 

35.  Is  f  contained  in  $  ? 

36.  Is  -J  contained  in  2$  ? 

37.  Is  £  contained  in  9  ? 


237.  Principle. — To  divide  by  a  fraction,  multiply 
by  the  denominator,  and  divide  the  product  by  the  numer- 
ator. 

WRITTEN     EXERCISES. 

238.  1.  Divide  180  by  A- 

operation.  Analysis.— Multiply  180  by 

■1  dq_j_  8  180  X  15—8  =  3374     the  denominator  15,  and  divide 

the  product  by  the  numerator 
8  (Prin.  ),  which  is  equivalent  to  multiplying  180  by  the  reciprocal 
ofTS,oTi£.    (197.) 

In  like  manner  divide 

2.  63  by  ^.  4.     120  by  ^.  6.     316  by  ^. 

3.  96  by  if  5.     276  by  ff.  7.     604  by  £f- 

8.  If  f  of  an  acre  of  land  cost  $63,  what  cost  1  acre  ? 

9.  Divide  84  by  6|?  11.  Divide  1260  by  42f 
10.  Divide  195  by  9^.  12.  Divide  2400  by  35|. 

13.  Paid  i  of  $64  for  \  of  17£  cords  of  wood.  What 
was  the  cost  a  cord  ? 

14.  A  man  gave  503  acres  of  land  to  his  sons,  giving 
them  83|  acres  apiece.    How  many  sons  had  he  ? 


130 


FKACTIOKS. 


15.  Divide  ^  by -f. 


OPERATION. 


Analysis. — &  divided  by  |, 
is  equal  to  36  fortieths  divided 
by  15  fortieths,  which  gives  2|, 
the  required  quotient.     Or, 
Since  1  divided  by  f  is  f,  ^ 
of  1  divided  by  f  is  ^  of  f ,  or  f  §  equal  to  2|,  the  same  result. 

It  is  obvious  that  the  result  is  obtained  by  multiplying  the  nu- 
merator of  the  dividend  by  the  denominator  of  the  divisor,  and  the 
denominator  of  the  dividend  by  the  numerator  of  the  divisor. 
Hence  by  inverting  the  terms  of  the  divisor  and  using  its  reciprocal 
(197),  the  operation  becomes  the  same  as  multiplying  one  fraction 
by  another. 

16.  Divide  |  of  |  by  J  of  ^. 

operation.  Or,  Analysis. — The   divi- 

dend reduced  to  its  sim- 
plest form  is  ^  ;  the  divi- 
sor reduced  in  like  man- 
ner is  T5F,  and  i  divided 
by  T5¥  is  1§,  the  quotient 
required.     Or, 

Invert  both  factors  of 
the  divisor,   and    obtain 
the  result  by  a  single  operation. 

If  the  vertical  line  is  used,  the  numerators  of  the  dividend  and 
the  denominators  of  the  divisor  are  written  on  the  right. 

In  like  manner 

17.  Divide  \  by  f  20.  Divide  J  of  \\  by  f  of  f 

18.  Divide  ^  hJ  I-  21-  Divide  3i  hJ  i  of  2i- 

19.  Divide  f  by  ff .  •  22.  Divide  f  of  2J  by  J  of  3. 

Hence  the  following  general 

Kule. — Reduce  the  dividend  and  divisor  to  fractions 
having  a  common  denominator,  then  divide  the  numerator 
of  the  dividend  by  the  numerator  of  the  divisor.    Or, 


Or, 

5 

3 

0 

0 

H 

0 

$ 

14* 

5 

6 

n 

DIVISION.  131 

I.  Reduce  the  integers  and  mixed  numbers,  if  any,  to 
improper  fractions. 

II.  Multiply  the  dividend  by  the  reciprocal  of  the  divisor. 
Apply  cancellation  when  practicable. 

Divide 


23.     56  by  If 

26. 

Abyii. 

29. 

16|byl3i. 

24.  HI  by  80. 

27. 

45J-  by  8. 

30. 

^  of  4  by  |  of  3J. 

25.  l«by«. 

28. 

92  by  5f 

31. 

44  by  f  of  5£  x  7. 

32.  What  number  multiplied  by  f  will  produce  912-J-g-? 

33.  Of  what  number  is  52 J  the  £  part  ? 

34.  What  number  multiplied  by  33|  will  produce  297J-  ? 

35.  If  |  of  a  farm  cost  $6270,  what  did  the  whole  cost  ? 

36.  If  14  acres  of  meadow  yield  32|  tons  of  hay,  how 
much  do  5  J  acres  produce  ? 

37.  If  7f  yards  of  velvet  are  worth  $17£,  what  is  J  of  a 
yard  worth  ? 

38.  What  will  be  the  cost  of  24J  pounds  of  sugar,  if  3| 
pounds  cost  $.33  ? 

39.  What  is  the  value  of  -j^  ? 

6f 

operation.  Analysis.  —  This  example 

10^       fLL  is  only  another  form  for  ex- 

~g]T  =  ]TI  ~~    »     •   "**>  a  pressing  division  of  fractions. 

3  Hence,    after    reducing    the 

^  ~^~  "¥"  —  "¥"  X  "A:  =  f  —  li     mixed  numbers  to  improper 
2  fractions,  treat  the  numerator 

-\i  as  a  dividend,  and  the  denominator  ^-  as  a  divisor,  then  pro- 
ceed according  to  the  rule  for  the  division  of  fractions. 

Expressions  similar  to  the  above  are  sometimes  called  Complex 
Fractions,  and  the  process  of  performing  the  division  is  called 
reducing  a  complex  fraction  to  a  simple  one. 

If  either  the  numerator  or  the  denominator  consists  of  one  or 
more  parts  connected  by  +  or  — ,  the  operations  indicated  by  these 
signs  must  first  be  performed,  then  the  division. 


132 


FRACTIONS. 


40.  What  is  the  value  of  |,  or  of  -J  -?- }  ? 

4 

41.  What  is  the  value  of  ^f,  or  of  12£  ^- 10|  ? 
42„  What  is  the  value  of  ^13,  or  of  117£  ~-  18  ? 

43.  What  is  the  value  of  *^l?A,^rof  f  x  2^-^? 

TO 

44.  Find  the  value  of  \°  .%,  or  of  f  x  H  -*-  tV  x  5f 

45.  Find  the  value  of  ?  J"-L  or  of  }  — |  -=-  £  -f  -}. 

3"    i     ¥ 

46.  If  a  man  spend  $4f  a  month  for  tobacco,  in  what 
time  will  he  spend  $27£  ? 

Find  the  value 


47.  of  »*i±a, 

48.  Of  f-±^|. 
If  —  i 


49.  Of  |  x  *  -T-  6*  -  5^. 

50.  Of  (16|  4-  18J)  x  17. 

5LOf9}x8xA-f  «. 
52.  Of(7A-5TV)-(4i  +  fi|). 


EELATION    OF    NUMBEES. 

239.  Numbers  to  be  compared  with  each  other,  must 
be  so  far  of  the  same  nature,  that  one  may  properly  be 
said  to  be  a  part  of  the  other. 

Thus,  we  may  compare  a  day  with  a  week,  since  the  one  is  the 
seventh  part  of  the  other ;  but  we  cannot  say,  that  a  day  is  any 
part  of  a  mile,  therefore  a  day  cannot  be  compared  with  a  mile. 

240.  Principle. — Only  like  numbers  are  so  related 
as  to  be  compared  with  each  other. 


RELATION     OF     NUMBERS.  133 

241.  To  find  what  part  one  number  is  of  another. 

ORAL    EXERCISES. 

1.  What  part  of  5  is  3  ? 

Analysis.— Since  1  is  £  of  5,  3  is  3  times  \  or  §  of  5  ;  or  it  is  3 
divided  by  5.     Hence  3  is  f-  of  5. 

2.  What  part  of  9  is  5  ?     Of  12  is  7  ?     Of  24  is  18  ? 

3.  10  yards  are  what  part  of  25  yards  ?  8  pounds,  of 
20  pounds  ?   9  eggs,  of  a  dozen  ?  10  ounces,  of  a  pound  ? 

4.  $15  are  what  part  of  $50  ?  $60,  of  $72  ?  9  days, 
of  90  days  ?     6  days,  of  a  week  ?     7  months,  of  a  year  ? 

5.  If  an  acre  of  land  can  be  bought  for  $48,  what  part 
of  an  acre  can  be  bought  for  $8  ?    For  $12  ?   $16?   $24? 

6.  What  part  of  3  is  f  ? 

Analysis. — 1  is  ^  of  3,  and  f  of  1  is  f  of  \  of  3,  or  |  x  f  —  ^-. 
Or,  3  =  V  ;  tlie  relation  of  -8B±  to  f  is  the  same  as  that  of  their 
numerators  24  and  5,  or  ¥5T.    Hence  f  is  -^  of  3. 

7.  What  part  of  9  is  J-  ?     Of  8  is  ^?     0f  20  is  |  ? 

8.  What  part  of  15  is  1J  ?     Of  18  is  2-J  ?    Of  25  is  6J  ? 

9.  |  of  a  month  is  what  part  of  8  months  ? 

10.  What  part  of  £  is  f  ? 

Analysis.— 1  fifth  is  |  of  4  fifths,  and  1,  or  5  fifths,  is  5  times  \ 
or  |  ;  nence  f  is  f  of  f,  or  ||,  equal  to  f. 

Or,  £  =  f§,  and  §  =  |f ,  and  the  relation  of  f  to  |  is  the  same  as 
that  of  10  to  12,  or  \%  =  f.     Hence  f  is  \  of  f. 

11.  What  part  of  £  is  \  ?     Of  -ft-  is  f  ?     Of  £  is  \  ? 

12.  What  part  of  ^  is  |?     Of  Jf  is  ^  ?     Of-^-isl? 

13.  What  part  of  3£  is  f  ?     Of  4£  is  3  ?     Of  2|  is  If  ? 

14.  What  part  of  7£  is  1J  ?    Of  If  is  ft  ?    Of  3  \  is  2}  ? 

15.  What  part  of  9  miles  are  f  of  8  miles  ?  J  of  10  miles  ? 


134 


FRACTIONS. 


WRITTEN     EXERCISES 


242.  What  part  of 


1.  96  is  72  ? 

2.  56  is  |  ? 

3.  120  is  90? 

4.  H^  If? 


9.  150  is  12|? 

10.  24f  is  rt? 

11.  160is26f? 

12.  212J- is  42f  ? 


5.  ***** 

6.  6  is  |f? 

7.  80is5i? 

8.  13}  is  2^? 

13.  A  man  having  $150,  gave  $25  for  a  robe,  and  f  of  the 
remainder  for  a  harness.     What  part  of  $150  had  he  left? 

14.  Bought  a  horse  for  $275,  and  sold  him  for  $160. 
For  what  part  of  the  cost  was  he  sold  ? 

15.  If  from  18}  yards  of  cloth  2$  yards  are  cut,  what 
part  of  the  whole  is  taken  ? 

16.  If  15  tons  of  coal  cost  $112£,  what  part  of  $112J  will 
}  of  a  ton  cost  ? 

243.  To  find  a  number  when  a  fractional  part  of 
it  is  given. 

ORAIj    exercises. 

1.  7  is  -J-  of  what  number. 

Analysis. — 7  is  |  of  5  times  7,  which  is  35.    Hence  7  is  \  of  35. 

2.  12  is  {  of  what  number  ?    \  of  what  number  ? 

3.  9£  is  \  of  what  number  ?■'    J  of  what  number  ? 

4.  7f  is  ^  of  what  number  ?    -^  of  what  number  ? 

5.  36  is  }  of  what  number  ? 

Analysis. — Since  36  is  -J  of  a  certain  number,  ]  of  the  number  la 
I  of  38,  or  12  ;  and  the  number  is  4  times  12,  or  48. 

6.  42  is  |  of  what  number  ?    -^  of  what  number  ? 

7.  75  is  4  of  what  number  ?    }  of  what  number  ? 

8.  84  is  ff  of  what  number  ?    TV  of  what  number  ? 

9.  15}  is  $  of  what  number  ?     ^  of  what  number  ? 


II  EL  ATI  OK     OF     IS' UMBERS.  135 

10.  |  is  f  of  what  number  ?    £  of  what  number  ? 

11.  If  is  -J  of  what  number  ?    -|  of  what  number? 

12.  3f  is  ^  of  what  number  ?    £  of  what  number  ? 

13.  36  is  |  of  how  many  times  4  ? 

Analysis.— 36  is  f  of  8  times  £  of  36  which  is  32,  and  4  is  con= 
fcained  in  32,  8  times.    Hence  36  is  f  of  8  times  4. 

14.  28  is  -j*.  of  how  many  times  8  ?     12  ?     9  ?     16  ? 

15.  35  is  i  of  how  many  times  |  of  28  ?    -J-  of  30  ? 

•     16.  16|  is  J  of  how  many  times  \  of  56  ?    -j»fr0f48? 

17.  |  is  ^  of  how  many  times  \  of  \  ?    J  of  |  ? 

18.  |  of  56  is  ^g-  of  what  number? 

Analysis. — |  of  56  is  3  times  |  of  56,  which  is  21  ;  and  21  is  ^ 
of  10  times  }  of  21,  which  is  30.    Hence  f  of  56  is  TV  of  30. 

19.  $  of  27  is  f  of  what  number  ?    -f  of  what  number  ? 

20.  f  of  f  of  64  is  f  of  what  number  ? 

21.  {■  of  ^  of  72  is  \  of  |  of  what  number  ? 

22.  |  of  54  is  |  of  how  many  times  5  ?     7  ?    8  ?    9  ? 

23.  f  of  -J  of  63  is  J  of  |  of  how  many  times  10  ?    9  ? 

24.  4-  of  56  is  $  of  3  times  what  number  ? 

Analysis.— ^  of  56  is  32,  and  32  is  §  of  36,  and  36  is  3  times  £  of 
36,  which  is  12.     Hence  f  of  56  is  §  of  3  times  12. 

25.  f  of  64  is  £  of  9  times  what  number  ? 

26.  £  of  21  is  J  of  8  times  what  number  ? 

27.  Paid  $60  for  a  sideboard)  which  was  -f  of  the  cost 
of  a  bookcase.    What  was  the  cost  of  the  bookcase  ? 

28.  A  scarf  cost  $lf,  which  was  £  of  the  cost  of  a  vest. 
What  was  the  cost  of  the  vest  ? 

29.  Paid  $100  for  a  sleigh,  which  was  f  of  3  times  what 
I  paid  for  a  harness.     What  did  I  pay  for  the  harness  ? 

Written  Exercises  of  this  kind  are  included  in  the  review  examples. 


136  FRACTIONS. 

REVIEW    OF   FRACTIONS. 

ORAL      EXAMPLES. 

244.  1.  What  fraction  added  to  -|  will  make  £  ? 

2.  What  number  taken  from  25f  will  leave  7f  ? 

3.  If  the  sum  of  two  fractions  is  J-J  and  one  of  them  is 
J-,  what  is  the  other? 

4.  From  what  number  must  3f  be  taken  to  leave  5-J-  ? 

5.  A  boy  spends  f  of  his  earnings  for  board,  and  J  for 
clothing.     What  part  has  he  left  ? 

6.  The  less  of  two  numbers  is  5TV,  and  their  difference 
J.     What  is  the  greater  ? 

7.  What  number  divided  by  £  will  give  a  quotient  of  If  ? 

8.  The  product  of  two  numbers  is  4,  and  one  of  them 
is  18.     What  is  the  other? 

9.  If  2  be  added  to  both  terms  of  the  fraction  f ,  will  its 
value  be  increased,  or  diminished,  and  how  much  ? 

10.  If  2  be  added  to  both  terms  of  the  fraction  -J,  will 
its  value  be  increased,  or  diminished,  and  how  much  ? 

11.  If  a  box  of  tea  cost  $21-f,  what  will  f  of  a  box  cost  ? 

12.  A  man  owning  |  of  a  steam-mill  sold  £  of  his  share. 
What  part  of  the  whole  mill  does  he  still  own  ? 

13.  A  farmer  sold  40  acres  of  land,  which  was  ^  of  his 
whole  farm.     How  many  acres  were  there  in  his  farm  ? 

14.  A  man  sold  {  of  his  farm,  and  had  100  acres  left. 
How  many  acres  had  he  at  first  ? 

15.  Bought  a  watch  and  chain  for  $120,  the  chain  cost- 
ing -f  as  much  as  the  watch.     What  did  each  cost  ? 

16.  A,  B,  and  C  together  own  a  yacht.     A  owns  $  of 
it,  and  B,  $  of  it.     What  part  does  C  own  ? 


REVIEW.  137 

17.  A  farmer  put  all  his  grain  into  4  bins  :  in  the  first 
he  put  |-  of  it,  in  the  second  J,  in  the  third  |j  and  in  the 
fourth  40  bushels.     How  many  bushels  of  grain  had  he  ? 

18.  Bought  6  mats  at  $f  each,  and  had  $5  left.  How 
much  money  had  I  at  first  ? 

19.  How  many  bushels  of  grain  can  be  put  into  15  bags, 
if  they  hold  2|  bushels  each  ? 

20.  If  5  men  can  do  a  piece  of  work  in  10$  days,  how 
many  days  will  it  take  one  man  to  do  the  same  ? 

21.  If  a  man  can  build  6  rods  of  wall  in  1  day,  how 
many  rods  can  he  build  in  7f  days  ? 

22.  How  much  less  than  $10  will  7  pounds  of  tea  cost, 
at  $|  a  pound  ? 

23.  George  having  $1-|,  gave  f  of  it  for  a  knife.  What 
part  of  a  dollar  did  he  give  for  his  knife  ? 

24.  At  $12 \  a  ton,  what  will  f  of  a  ton  of  hay  cost  ? 

25.  Bought  a  cow  for  $45 J,  and  sold  her  for  -fa  of  what 
she  cost.     What  did  I  lose  ? 

26.  If  a  man  has  22f  bushels  of  clover-seed,  and  he 
sells  J  of  it,  how  much  has  he  left? 

27.  What  will  4J  days'  wages  come  to  at  $2 J-  a  day? 

28.  A  man  spent  f  of  his  money,  and  then  found  that 
$15  was  |  of  what  he  had  left.     What  had  he  at  first  ? 

29.  A  man  paid  $30  for  a  cow,  f  of  the  cost  of  which 
was  |  of  the  cost  of  a  horse.     What  did  the  horse  cost  ? 

30.  How  many  pounds  of  tea  worth  $T^  a  pound,  must 
be  given  for  9  bushels  of  apples  worth  $£  a  bushel  ? 

31.  How  many  building  lots  of  -^  of  an  acre  each  are 
contained  in  1£  acres  o£  land  ? 

32.  At  $  J  each,  how  many  books  can  be  bought  for  $3^  ? 

33.  If  |  of  a  box  of  figs  cost  $1-|,  what  will  1  box  cost  ? 


138  FRACTIONS. 

34.  If  5-J-  dozens  of  eggs  cost  $1$,  what  is  the  cost  of  1 
dozen  ?    Of  %\  dozens  ?    Of  Z\  dozens  ? 

35.  If  \  of  a  barrel  of  flour  cost  $8,  what  cost  9  barrels  ? 

36.  If  3  yards  of  flannel  cost  $|,  what  will  8  yards  cost  ? 

37.  How  much  tea  can  be  bought  for  $4J,  at  $f  a  pound  ? 

38.  If  f  of  a  bushel  of  quinces  cost  $-§-,  what  will  1 
bushel  cost  ?    2-J-  bushels  ?    3}  bushels  ? 

39.  If  a  gallon  of  syrup  cost  $£,  how  many  gallons  can 
be  bought  for  |&  ?    For  $1|  ?    Forlf? 

40.  A  man  having  $24,  gave  f  of  his  money  for  clover- 
seed  at  I5-J-  a  bushel.     How  many  bushels  did  he  buy  ? 

41.  What  number  taken  from  2  J  times  12f  leaves  20J  ? 

42.  A  coal  dealer  sold  f  of  what  coal  he  had  on  hand 
for  $90,  at  the  rate  of  $6  a  ton.     How  many  tons  had  he  ? 


WRITTEN     EXAMPLES. 

24:5.  1.  Change  £of  f,  |,  -J,  and  %,  to  equivalent  frac- 
tions whose  denominator  shall  be  72. 

2.  Find  the  least  common  denominator  of  f ,  f ,  £,  and  1|-. 

3.  The  less  of  two  numbers  is  1206|  and  their  differ- 
ence 470£.     Find  the  greater  number. 

4.  Find  the  value  of  (3  x|x|  x4f)-(3|xf  x4xf). 

5.  What  number  multiplied  by  f  will  produce  1825-J  ? 

6.  What  number  diminished  by  %  and  £  of  itself  leaves 
a  remainder  of  144  ? 

7.  If  f  of  a  farm  is  valued  at  $1729 J,  what  is  the  value 
of  the  whole  ?  * 

8.  A  man  gave  -J,  $ ,  and  -J-  of  his  money  for  different 
objects  and  had  $1500  left.     How  much  had  he  at  first  ? 


REVIEW.  139 

9.  If  the  dividend  is  -J,  and  the  quotient  ^,  what  is  the 
divisor  ? 

10.  A  man  owning  |-  of  a  cotton  mill,  sold  §  of  his 
share  for  $4560f    What  was  the  value  of  the  mill  ? 

11.  A  stone  mason  worked  23  J  days,  and  after  paying 
f-  of  his  earnings  for  board  and  other  expenses,  had  $53  J- 
left.     What  did  he  receive  a  day  ? 

12.  Gave  6§  pounds  of  butter  at  36  cents  a  pound,  for 
3£  gallons  of  oil.    What  was  the  oil  worth  a  gallon  ? 

13.  A  person  having  2 71 J  acres  of  land,  sold  J  of  it 
to  one  man,  and  -f  of  it  to  another.  What  was  the  value 
of  the  remainder  at  $57£  an  acre  ? 

14.  A  man's  family  expenses  are  $2465^  a  year,  which 
is  f  of  his  income.    What  does  he  save  ? 

15.  If  7£  tons  of  hay  cost  $120,  how  many  tons  can  be 
bought  for  $78  ? 

10.  If  a  man  travel  240  miles  in  5f  days,  how  far  would 
he  travel  in  3|  days  ? 

17.  A  can  do  a  certain  piece  of  work  in  8  days,  and  B 
can  do  it  in  6  days  :  in  what  time  can  both  together  do  it  ? 

18.  A,  B,  and  C  can  do  a  piece  of  work  in  5  days ;  B 
and  0  can  do  it  in  8  days  :  in  what  time  can  A  do  it  alone  ? 

19.  If  f  of  4  acres  of  land  cost  $205f,  what  will  £  of  2 
acres  cost  ? 

20.  Bought  i  of  25J  yards  of  cloth  for  £  of  $177f 
What  was  the  cost  per  yard  ? 

21.  If  |  of  a  farm  is  worth  $9000,  what  is  -&  of  it  worth  ? 

22.  If  8  be  added  to  both  terms  of  the  fraction  f|-,  will 
its  value  be  increased,^  diminished,  and  how  much  ? 

23.  If  8  be  added  to  both  terms  of  the  fraction  *£>  will 
its  value  be  increased,  or  diminished,  and  how  much  ? 


140  FRACTIONS. 

24.  How  many  bushels  of  oats  at  $  J  a  bushel,  will  pay 
for  |  of  a  barrel  of  flour  at  $9-|-  a  barrel  ? 

25.  A  man  at  his  death  left  his  wife  $12500,  which  was 
J  of  |  of  his  estate.  At  her  death  she  left  \  of  her  share 
to  her  daughter.  What  part  of  the  father's  estate  did  the 
daughter  receive  from  her  mother  ? 

26.  Paid  $1837£  for  3675  bushels  of  oats.  What  was 
the  cost  a  bushel  ? 

27.  A  merchant  bought  a  cargo  of  flour  for  $21 73 J,  and 
sold  it  for  f-f  of  the  cost,  thereby  losing  $.25  on  a  barrel. 
How  many  barrels  of  flour  did  he  purchase  ? 

28.  A  man  owning  \  of  156f  acres  of  land,  sold  \  of  £ 
of  his  share.  How  many  acres  did  he  sell,  and  what  was 
the  value  of  the  remainder  of  his  share,  at  $42|  an  acre  ? 

29/  A  horse  and  wagon  cost  $360 ;  the  horse  cost  2-J- 
times  as  much  as  the  wagon.    Find  the  cost  of  the  wagon  ? 

30.  If  $7£  will  buy  3£  cords  of  wood,  how  many  cords 
can  be  bought  for  $31  \  ? 

31.  A  dealer  sold  7  barrels  of  apples  for  $32£,  which 
was  \  as  much  as  he  received  for  all  he  had  left,  at  $4  a 
barrel.     How  many  barrels  in  all  did  he  sell  ? 

32.  If  f  of  9  bushels  of  wheat  cost  $13|,  what  will  -J  of 
a  bushel  cost  ? 

33.  A  man  engaging  in  trade  lost  f  of  the  money  he  in- 
vested, after  which  he  gained  $740,  and  then,  had  $3500. 
What  was  his  loss  ? 

34.  A  boy  having  lost  -J-  of  his  kite-string,  added  45} 
feet ;  the  string  was  then  %  of  its  original  length.  What 
was  its  original  length  ? 

35.  There  are  two  numbers  the  sum  of  which  is  4J,and 
their  difference  \ .    What  are  the  numbers  ? 


REVIEW. 


141 


36.  A  man  invests  £  of  his  money  in  cotton,  J  in  sugar, 
■fg  in  molasses,  and  the  remainder,  which  is  $2542,  in  dried 
fruits.  What  is  the  amount  of  each  investment,  and  the 
total  amount? 

37.  If  f|  of  3 J  times  1,  be  multiplied  by  -J,  the  pro- 
duct divided  by  f,  the  quotient  increased  by  4J-,  and  the 
sum  diminished  by  f  of  itself,  what  is  the  remainder  ? 

Reduce  to  their  simplest  form  : 

t0f:f 


38. 


**  +  * 


*  3t4^- 


40. 


41. 


f°H 


|of  74, 

|  of  15   '    11  x  If 


+ 


I  of  4 


Complete  the  following  equations  : 


24-fofl$ 


"ai+tt"1 


!4/   "   8# 


1  + 


43. 


3»  +  2|      3|  x34 
34-2| +  3J-^V 


H 

7 


45. 


!-* 

1+1 

1-1 

44-     (H+ltf^8f  +  3«)--(3*+tt-S-3l  +  14A)=? 


246, 


SYNOPSIS  FOR  REVIEW. 


FRACTIONS. 


1.  Equal  Parts. 

2.  Principles,  1  and  2. 

3.  Definitions. 


1.  A  Fraction. 

2.  A  Fractional  Unit. 


4.  Expression  op  Fractions. 

.  Terms.  \  L  Denominator. 

Numerator. 


\i 


142 


FRACTIONS 


SYNOPSIS    FOR    REVIEW— Continued. 


1.  Classification 


2.  Definitions. 


•1 


1.  Proper  Fractions. 

2.  Improper  Fractions. 

1.  Mixed  Numbers. 

2.  Reciprocal  of  a  Number. 


8. 


Fraction. 


3.  Value  of  a  Fraction. 

4.  General  Principles,  1,  2,  3. 

5.  General  Law. 


J   1.  Reduction. 

1    T)  f     J  ^  Hiyher  Terms. 

'   I  3.  Lower  Terms. 

202. 

t  4.  Lowest  Terms. 

2.  Principle. 

^  3.  Rules,  1,  2. 

210. 

Rule. 

G. 

Reduction.  < 

212. 

Rule. 

f  1.  Common   Denomi- 

214. 

J   2.  Z^«5i  Common  De- 
[       nominator. 
2.  Principles,  1,  2. 

>. 

L  3.  Rule  (1) ;  I,  II.     (2.) 

7. 

Addition.             j      . 

'rinciple. 
lule,  I,  II. 

8. 

Subtraction 

(1.1 
'        (2.  I 

'rinciple. 
tule,  I,  II. 

{  227.  Principles,  1.  2.  {Rule 1, 11(1) 
9.  Multiplication,  j  23Q   Principle  }  j^  7>  J7  (3> 

10.  Division.  ffiS^}^"» 

(  241. 

11.  Relation  of  Numbers.       Principle,  j  242. 


ORAE    exercises. 

247.  1.  If  a  unit  be  divided  into  10  equal  parts,  what 
is  each  part  called  ?    What  are  2  parts  ?  3  parts  ?  4  parts  ? 

2.  What  is  the  fractional  unit  ? 

3.  If  1  tenth  of  a  unit  be  divided  into  10  equal  parts, 
what  is  each  part  called  ?  What  are  2  parts  ?  4  parts  ? 
5  parts  ?     7  parts  ?     12  parts  ?    25  parts  ? 

4.  What  is -A- of  ^  ?    A  of  A?    tWA?    A  of  A? 

5.  If  a  unit  be  divided  into  100  equal  parts,  or  each 
tentli  into  10  equal  parts,  what  are  the  parts  called  ? 


144  DECIMALS. 

6.  What  part  of  1  tenth  is  1  hundredth  ?  How  many 
hundredths  in  1  tenth  ? 

7.  If  1  hundredth  of  a  unit  be  divided  into  10  equal 
parts,  what  is  each  part  called?  What  are  3  parts? 
What  are  8  parts  ?     9  parts  ?     15  parts  ? 

-8.  What  is  ^  of  ^  of  ^?    iVofyfe?    Aof^? 

9.  If  a  unit  be  divided  into  1000  equal  parts,  or  each 
hundredth  into  10  equal  parts,  what  ar.  the  parts  called? 
What  are  12  parts  ?     26  parts  ?    42  parts  ? 

10.  What  part  of  1  hundredth  is  1  thousandth  ?  How 
many  1  thousandths  is  1  hundredth  ? 

NOTATION    AND    NUMEBATION. 

24:8,  A  Decimal  Fraction  is  one  or  more  of  the 
decimal  divisions  of  a  unit.  Thus,  -fa,  ^ ,  y^,  -f  f&ij-j  etc, 
are  decimal  fractions. 

Decimal  Fractions  are  commonly  called  Decimals*    (16.) 

249.  Decimals  are  like  other  Fractions,  except  that 
their  denominators  increase  and  decrease  by  the  uniform 
scale  of  10.  The  fractional  units  are,  therefore,  always 
tenths,  hundredths,  thousands,  etc. 

250.  The  Decimal  Sign  ( . ),  called  the  decimal 
point,  is  used  to  distinguish  a  decimal  from  an  integer, 
and  must  ahvays  be  placed  before  the  numerator  of  the 
decimal. 

*  The  terms  fraction  and  decimal  will  hereafter  be  used  to  dis- 
tinguish the  common  from  the  decimal  form  of  expression.  Thus, 
y7^,  and  .75,  are  two  forms  of  expressing  the  same  thing.  For 
convenience  we  shall  call  the  first  form  a  fraction,  and  the  other  a 
decimal. 


NOTATION     AND     NUMERATION.  145 

251,  The  position  of  the  decimal  sign  indicates  the 
denominator,  and  determines  the  value  of  the  decimal 
expression.     Thus, 


-^  is  expressed  .7. 
f3A "        "         -36. 


■j^q  is  expressed  .126. 
'iWfV "        'l         .1425. 


252.  The  Denominator  of  a  decimal  fraction  is 
always  10,  100,  1000,  etc.,  or  1  with  as  many  ciphers 
annexed  as  therf  ^are  figures  in  the  given  decimal.  Thus, 
A  =  Ar;  .09  =  ^;  .007  =  T ^,  etc. 

253.  The  Numerator  of  a  decimal  fraction  when 
expressed  alone,  must  have  as  many  decimal  places  as 
there  are  ciphers  in  the  denominator.  Thus,  -fa  =  .8  ; 
tWt  =  .12;  ^=.125,  etc. 

If  the  numerator  does  not  contain  as  many  figures  as  there  are 
ciphers  in  the  denominator,  prefix  ciphers  until  the  number  of 
places  is  equal  to  the  number  of  ciphers  in  the  denominator,  and 
prefix  the  decimal  point.     Thus,  T£¥  ==  .07  ;   TQ%^  =  .009,  etc. 

254:.  Decimal  fractions  may  be  written  in  two  ways  ; 
either  as  other  fractions,  the  denominator  being  expressed, 
or,  in  decimal  notation,  the  denominator  being  omitted. 
Thus, 

■j^y,   or  .5    is  read  5  tenths  and  is  ^  of  5  units. 

yjb-,    ".05      "      5 hundredths,     «    -^"5 tenths. 
•nfVo,  ".005    "      5 thousandths,    "    ^  "  5 hundredths. 

255.  The  value  of  any  decimal  figure  is  always  ^  of 
the  value  of  the  same  figure  in  the  next  place  to  the  left. 

256.  When  an  integer  and  decimal  are  written  to- 
gether, the  expression  is  a  Mixed  Number  (195).  Thus, 
7.12  and  26.134  are  mixed  numbers. 

251.  The  relation  of  decimals  and  integers  to  each 
other  is  clearly  shown  by  the  following 
7 


146 

DECIMALS. 

Table. 

CO 

. 

GO 

4 

9 

5 

CO 

n3 

1 

pi 

0 

co 

CO 

rS 

PI 

.2 

S 
1 

*d 

.2 

GO 

.2 
1 

00 

OQ 

o 

s 

CO 

Pi 
O 

03 
=0 

«Q 

g 

B:i 

0Q 

ft 

h3 

CD 

§ 

CO 

a 

0 
1— i 

=»3 

NO 

PI 

.2 
1 

pf 

w 

i 

l-o> 
t~o 

Si 

PI 

m 

pi 
p 

GO 

a. 

Q 

£          g 

PI 
PS 

'W 

5S 

pi 
H 

pi 

S3 

r~o 

1 
a 

9 

8 

7 

6 

5 

4 

3 

2 

1  .2 

3 

4 

5 

6 

7 

8 

9 

Integers. 

V 

Decimals. 

The  number  is  read  987  million  654  thousand  321,  awrf 
23  million  456  thousand  789  hundred-millionths. 
A  decimal  takes  the  name  of  its  right-hand  order. 

258.  In  decimals,  as  in  integers,  make  the  order  of 
units  the  starting-point  of  notation  and  of  numeration, 
extending  the  scale  to  the  left  of  the  units'  place  in 
writing  integers,  and  to  the  right  of  the  units'  place  in 
writing  decimals. 

The  first  order  to  the  left  of  units  is  tens,  and  the  first  order  to 
the  right  of  units  is  tenths  ;  the  second  order  to  the  left  of  units  is 
hundreds,  and  the  second  order  to  the  right  is  hundredths ;  the 
third  order  to  the  left  is  thousands,  and  the  third  order  to  the  right 
is  thousandths,  and  so  on,  the  integers  on  the  left,  and  the  decimals 
on  the  right,  equally  distant  from  the  units'  place,  corresponding 
in  name. 

259.  Hence,  both  in  integers  and  in  decimals,  the 
value  of  any  figure  is  determined  by  the  position  of  that 
figure,  and  is  always  ten  times  the  value  of  the  same  figure 
in  the  next  lower  order,  or  1  tenth  the  value  of  the  same 
figure  in  the  next  higher  order.     Hence, 

260.  In  writing  decimals,  vacant  orders  must  be  filled 
with  ciphers.     (36,  2.) 


NOTATION     AND     NUMERATION 


147 


Dictation  exercises,  both  oral  and  written,  should  be  given,  until 
the  pupil  can  write  and  read  decimals  with  rapidity  and  correctness. 
Orai,  thus,  Ques.  "  The  denominator  of  a  fraction  is  100,  the  nu- 
merator 7  ;  what  will  express  the  decimal  ?  "  The  prompt  response 
should  be,  "Point,  naught,  seven,  read,  seven-hundredths "  (.07). 
Ques.  "  The  denominator  is  1000,  the  numerator  35."  Ans.  "Point, 
naught,  three,  Jive,  read  thirty-Jim  thousandths"  (.035),  etc. 

Also  the  converse ;  thus,  Ques.  *  *  Point,  naught,  eight ;  what  will 
express  the  fraction  ? "  Ans.  "  The  numerator  is  eight,  the  denom- 
inator one  hundred,  and  the  fraction  is  eight-hundredths"  (Tou). 
Ques.  "  Point,  naught,  one,  five  ?  "  Ans.  "The  numerator  is  fifteen, 
the  denominator  is  one  thousand,  and  the  fraction  fifteen-thou- 
sandths "  (To2tf),  etc. 


WRITTEN     EXERCISES. 


261.  Express  in  the  form  of  a  fraction, 


12. 

.16. 


.138. 
.003. 


,2162, 
.0056. 


Express  in  the  form  of  a  decimal, 


9. 
10. 


SB 

Too* 
Tooo 


11. 

12. 


ttfVo"' 

T&oT- 


13. 
14. 


3027 

toooo- 

3  09 

10000* 

7. 
8. 

15. 
16. 


.14036. 
.00935. 

1  0  0  0  0  o • 


262.  Prefixing  a  cipher  to  a  decimal  multiplies  the 
denominator  by  10,  and  hence  divides  the  decimal  by  10 
(200,  2).     Thus,  .5  =  &  ;  .05  =  ^ ;  .005  =  TJfUJF  ;  or, 

.5  _•_  io  =  .05  ;  .05  -*■  10  =  .005,  etc. 

263.  Rejecting  a  cipher  from  the  left  of  a  decimal 
divides  the  denominator  by  10,  and  hence  multiplies  the 
decimal  by  10  (200,  1).  Thus,  .007  =  -^  ;  .07 
=  ih>  ;  •?  =  A ;  or*  -0°7  x  10  =  .07  ;  .07  x  10  =  .7. 

264.  Annexing  a  cipher  to  a  decimal  multiplies  both 
numerator  and  denominator  by  10,  and  hence  reduces  the 
fraction  to  higher  terras  (200*  3).  Thus,  .3=-^-; 
.30  =  1%;  .300=-^. 


148 


DECIMALS. 


265,  Rejecting  a  cipher  from  the  right  of  a  decimal 
divides  both  numerator  and  denominator  by  10,  and  hence 
reduces  to  lower  terms  (200,  3).    Thus,  ^-5  =  .600 ; 

From  the  foregoing  explanations  are  deduced  the  fol- 
lowing 

266.  Pkikciples. — It  Decimals  are  governed  by  the 
same  laws  of  notation  as  integers.     Hence, 

2.  The  value  of  any  decimal  figure  depends  upon  the 
place  it  occupies  at  the  right  of  the  decimal  sign.     (258.) 

3.  Every  removal  of  a  decimal  figure  one  place  to  the 
right  diminishes  its  value  tenfold.     (262.) 

4.  Every  removal  of  a  decimal  figure  one  place  to  the  left 
increases  its  value  tenfold.     (263.) 

5.  Ciphers  may  be  annexed  or  rejected  at  the  right  of 
any  decimal,  without  changing  its  value.     (264,  265.) 

WRITTEN     EXERCISES. 


267.  Express  in  figures  and  decimally ; 
1.  Seventy-five  thousandths,    -^^hs  —  ^75. 


2.  Fifteen  hundredths. 

3.  Seven  thousandths. 

4.  Fifty-three  thousandths. 

5.  Nine  ten-thousandths. 


6.  22  ten-thousandths. 

7.  245  ten-thousandths. 

8.  1042  hundred-thousandths. 

9.  14605  millionths. 


10. 
11. 


tWo- 

7- 


a 
FoTJT' 


_3JLQ.5_ 
Toooo* 


12. 
13.  1 


5  72 
TO  0  0  0  0- 


14. 
15.  8 


16. 
17. 


10  0  0  0  o • 


Eule. — I.  Write  the  numerator  of  the  decimal  as  if  an 
integer,  writing  ciphers  in  the  place  of  vacant  orders  to 
give  each  significant  figure  its  proper  value,  and  place  the 
decimal  point  before  tenths. 


NOTATION     AND     NUMERATION. 


149 


II.  Read  the  decimal  as  if  an  integer,  and  give  it  the 
name  of  its  right-hand  order. 

In  like  manner  express  decimally  the  following  frac- 
tions and  mixed  numbers  : 


596  thousandths. 
625  ten-thousandths. 
12  ten  thousandths. 


21 


74  millionths. 
£2.  105  ten-millionths. 
23.  99010  billionths. 


18. 
19. 
20. 

24.  Four  hundred  thirty-seven  thousand  five  hundred 
49  millionths. 

25.  Three  million  forty  thousand  12  ten-millionths. 

26.  Six  hundred  and  24  hundred-million ths. 

27.  Four  hundred  ninety-five  million  seven  hundred 
five  thousand  and  43075  ten-millionths. 

28.  Four  million  seven  hundred  thirty-five  thousand 
and  903624  hundred-millionths.    . 


29. 

Toooo* 

33. 

205- 

unr- 

30. 

10000  0* 

34. 

68 

3  6 

uolooooo* 

31. 

10  0  3  5  4 
10  000  00* 

35. 

705^0^^ 

32. 

l  o  ootTooT* 

36. 

SOOnWMAnp 

Copy  and  read  the  following  decimals  and  mixed  num« 

bers  : 

37. 

.705. 

45. 

18.0031. 

53. 

.00078. 

38. 

.0023. 

46. 

6.306. 

54. 

.3050040. 

39. 

.3607. 

47. 

49.0703. 

55. 

.0003006. 

40. 

.00705. 

48. 

10.0064. 

56. 

42.0637. 

41. 

.400564. 

49. 

22.09042. 

57. 

108.0094. 

42. 

.000256. 

50. 

1.10106. 

58. 

230.40685. 

43. 

.0010275. 

51. 

14.00370. 

59. 

30.26002015. 

44. 

.0000407. 

52. 

70. 

00063. 

60. 

8.040103463. 

150  DECIMALS. 

DECIMAL    CUEEEXCY. 

268.  Currency  is  coin,  bank-bills,  treasury  notes, 
etc.,  employed  in  trade  and  commerce. 

269.  A  Decimal  Currency  is  a  currency  whose 
denominations  increase  and  decrease  by  the  decimal  scale. 

270.  The  Legal  Currency  of  the  United  States  is 
a  decimal  currency ;  it  is  sometimes  called  Federal  Money, 
because  issued  by  the  Federal  Government. 

Table. 


10  mills  (m.)                 make 

1  cent.     c. 

or  ct. 

10  cents                             " 

1  dime. 

d. 

10  dimes  or  100  cents      " 

1  dollar. 

i 

10  dollars                          " 

1  eagle. 

K 

271.  Since  the  dollar  is  the  unit  of  United  States 
Money,  dimes,  cents,  and  mills  are  respectively  tenths, 
hundredths,  and  thousandths  of  the  unit. 

272.  Dollars  should  be  written  as  integers,  with  the 
sign  ( $ ),  prefixed ;  and  dimes,  cents,  and  mills,  as  deci- 
mals; with  the  decimal  point  at  their  left,  or  before  tenths. 
Thus,  7  dollars  3  dimes  4  cents  5  mills,  are  written  $7,345. 

273.  The  denominations  eagles  and  dimes  are  not  re- 
garded in  business  operations,  eagles  being  tens  of  dollars, 
and  dimes  tens  of  cents.  Thus,  $34.27  is  read  34  dollars 
27  cents,  instead  of  3  eagles  4  dollars  2  dimes  7  cents. 

274.  Since  the  two  places  of  dimes  and  cents,  or  of 
tenths  and  hundredths  are  appropriated  to  cents,  when  the 
number  of  cents  is  less  than  10,  write  a  cipher  in  the 
place  of  tenths.     Thus,  9  cents  are  written  $.09.     (73.) 


SEDUCTION.  151 

275.  The  half-cent  may  be  written,  either  as  a  frac- 
tion (J),  or  as  5  mills.  Thus,  thirty-seven  and  a  half 
cents  are  written  $.37£,  or  $.375. 

276.  Cents  are  often  written  as  fractions  of  a  dollar. 
Thus,  $9.28  may  be  also  written  $9^-. 

277.  In  business  transactions,  if  the  mills  in  the  final 
result  are  5  or  more  than  5,  they  are  considered  a  cent,  if 
less  than  5,  they  are  not  regarded.  Thus,  $5,197,  would 
be  called  $5.20,  and  $5,194  would  be  called  $5.19. 

278.  Principles. — 1.  Decimal  currency  is  expressed 
according  to  the  decimal  system  of  notation. 

2.  All  the  operations  in  Decimal  Currency  are  the  same 
as  the  corresponding  operations  in  Decimals. 

EEDUCTIOJST    OF   DECIMALS. 

279.  To  reduce  decimals  to  units  of  lower  or 
higher  orders. 

ORAL    EXERCISES. 

1.  How  many  tenths  in  2  units  ?     In  5  units  ? 

2.  How  many  tenths  in  20  hundredths  ?     In  .40  ? 

3.  How  many  hundredths  in  2  units  ?    In  4  units  ? 

4.  How  many  hundredths  in  200  thousandths  ? 

5.  How  many  hundredths  in  5  tenths  ?  In  .6  ?  .7  ?  .8  ? 

6.  How  many  thousandths  in  .06  ?    Il  .25  ?   .48  ?  .75  ? 

7.  How  many  hundredths  in. 150?  In  .260?  In  .2500? 

8.  In  400  thousandths  how  many  hundredths  ?  Tenths  ? 

9.  How  many  tenths  of  a  dollar  in  $6  ?   Hundredths  ? 

10.  Change  4  dollars  50  cents  to  cents.    To  mills. 

11.  How  many  dollars  are  300  cents  ?     540  cents  ? 

12.  How  many  cents  are  2600  mills  ?    Dollars  ? 


152  DECIMALS. 

13.  What  is  the  decimal  expression  for  5  cents  ? 
An8.  Sign,  point,  naught,  five ;  ie&d  five  hundredths  ($.05). 

14.  Express  decimally  7  cents  ;  9  cents  ;  15  cents. 

15.  Express  decimally  7  mills  ;  5  cents  6  mills. 

16.  Express  decimally  2  dollars  45  cents  and  6  mills. 

Ans.  Sign,  two,  point,  four,  five,  six;  read,  two  and  four  hundred 
fifty-six  thousandths  dollars  ($2,456). 

17.  What  is  the  decimal  expression  for  84  cents  5  mills  ? 

18.  Change  .3  to  hundredths  ;  to  thousandths. 

19.  Change  .4  and  .05  to  thousandths  ;  .07  and  .01. 

20.  Change  .5,  .08,  and  .023  to  equivalent  decimals, 
having  a  common  denominator  of  1000.  Also,  .14,  .009, 
and  .6.     .7,  .007,  and  .091. 

21.  Eeduce  .7,  .150  and  .600,  to  equivalent  decimals, 
having  the  least  common  denominator.  Also,  .50,  .250, 
and  .1700.     .43,  .006,  and  .0214. 

280.  From  the  foregoing  it  appears, 

1.  That  dollars  may  be  reduced  to  cents  by  annexing 
two  ciphers  ;  and  to  mills,  by  annexing  three  ciphers. 

Omit  the  sign  %  and  write  cts.  or  m.  after  the  result. 

2.  That  cents  may  be  reduced  to  mills  by  annexing  one 
cipher. 

3.  That  cents  may  be  reduced  to  dollars  by  pointing 
off  two  figures  from  th#r  right ;  and  mills  to  dollars,  by 
pointing  off  three  figures  from  the  right,  and  prefixing 
the  sign  ($). 

4.  That  mills  may  be  reduced  to  cents  by  pointing  off 
one  figure  from  the  right. 

5.  That  two  or  more  decimals  are  reduced  to  a  common 
denominator  by  annexing  or  rejecting  ciphers  at  the  right 
until  the  decimal  places  of  all  are  equal. 


REDUCTION.  153 


WRITTEN     EXERCISES. 

281.  Reduce 

1.  $85  to  cents.   (280,  1. 

2.  $615  to  cents. 

3.  $24.06  to  cents. 

4.  $9,206  to  mills. 


5.  $57  to  mills. 

6.  86  cents  to  mills.  (280,2.) 

7.  $.763  to  mills. 

8.  $.47i-  to  mills. 


Change 
9.  486  cts.  to  dollars.  (280, 3.)  |  12.  8 

10.  32462  cents  to  dollars. 

11.  40327  mills  to  dollars. 


13.  50000  mills  to  dollars. 

14.  61040  cents  to  dollars. 


15.  Reduce  .7,  .05,  and  .304,  each  to  hundred-thou- 
sandths.    (280,  5.) 

16.  Reduce  2.5,  .107,  and .  0008,  each  to  ten-thousandths. 

17.  Change  4,  2.17,  .136,  and  .0408  to  equivalent  deci- 
mals having  a  common  denominator. 

18.  Reduce  9  tenths,  24  thousandths,  109  hundred- 
thousandths,  and  47  millionths  to  equivalent  decimals 
having  the  least  common  denominator.     Also, 

19.  100.03,  41.0034,  .475,  .0753,  and  6.00044. 

20.  .84003,  120.4,  5.00031,  and  15.240007. 

282.  To  reduce  a  deciinal*to  a  fraction. 
ORAIj     exercises. 

1.  How  many  halves  in  -&  ?    In  ^  ?    In  ^ff  ? 

2.  How  many  fifths  in  ^  ?     In  t^  ?    iWb  ?    -6  ? 

3.  How  many  fourths  in  ^fr?     In  .50?     In  .75? 

4.  How  many  twentieths  in  ^  ?     In  -^  ?    In  .20  ? 

5.  In  .50  how  many  halves  ?    Fourths  ?    Tenths  ? 


154 


DECIMALS 


WRITTEN     EXERCISES. 

283.  1.  Change  .375  to  an  equivalent  fraction. 

operation.  Analysis.— The  numerator  is  375,  the  de- 

o  «  k 3jjl_ £,     nominator  1000,  and  the  decimal  expressed 


as  a  fraction  is  TVw=t- 

Hence  375={ 

Change  to  equivalent  fractions, 

2.  .16. 

5.  $.75. 

8.  .024. 

11.  $.875. 

3.  .125. 

6.  $.375. 

9.  .5625. 

12.     .0008. 

4.  .625. 

7.  $.655. 

10.  .3125. 

13.     .9375. 

Eule. — Omit  the  decimal  point,  supply  the  proper  de- 
nominator, and  then  reduce  the  fraction  to  its  lowest  terms. 

14.  Eeduce  .13}  to  an  equivalent  fraction. 

,.,       131-       40        2 
°PERATI0N— 13^  =  ro^  =  300  =  l5- 

Reduce  to  fractions  in  their  lowest  terms, 


15.  $.37}. 

16.  $.62}. 

17.  $.08$. 


18.  .06}. 

19.  .58}. 

20.  .93|. 


21.  $.33}. 

22.  $.66f. 

23.  $.16|. 


Express  by  an  integer  and  a  fraction, 

27.  $15.4.       29.     $9,625.      31.  24.26f. 

28.  $36.75.     30.  $27,375.      32.  84.05$. 


24.  .1944*. 

25.  .444}. 

26.  .0008}. 

33.  38.41}. 

34.  104.00}. 


284.   To  reduce  a  fraction  to  a  decimal. 

ORAL     EXERCISES. 

1.  How  many  tenths  in  }?     How  many  hundredths  ? 
how  many  thousandths  ? 

2.  How  many  tenths  in  }  ?    Hundredths  in  }  ?    In  J  ? 

3.  How  many  hundredths  in  -fa  ?    In  fa  ?     In  -^  ? 


REDUCTION 


155 


WRITTEN    EXERCISES. 

285.  1.  Reduce  £  to  an  equivalent  decimal. 

operation.  Analysis. —Annex  the  same 

9  9.0.  —  J>J2  JL  —  -  G  2  5  number  of  ciphers  to  both  terms 


ooo 


_6J2  5_ 
1000 


of  the  fraction  and  divide  the  re- 
sulting terms  by  8,  the  significant  figure  of  the  denominator,  to 
obtain  the  decimal  denominator  1000.  Then  change  to  the  decimal 
form.    (253.) 

2.  Reduce  yf  3  to  an  equivalent  decimal. 

operation.  Analysis. — Since   yf^  = 

12  5)2.000  T3  5  of  2  units,  and  2  units 

equal  2000  thousandths,  y^g- 
•  01G        Or,  of    2000  t]wmandt}ls   is    ig 

if*  =  Tf  IH-o  =  T*ta  =.016       thousandths,  or  .016. 

Reduce  to  equivalent  decimals  : 

3.  $|.       I       5.     if.  7.     If- 
4-     «.       I       6.     }f       ;       8.     $^V. 

Rule. — I.  Annex  ciphers  to  the  numerator  and  divide 
by  the  denominator. 

II.  Point  off  as  many  decimal  places  in  the  result  as 

there  are  ciphers  annexed. 

The  sign  4-  is  sometimes  placed  after  the  result  to  indicate  that 
there  is  still  a  remainder.     Thus,  §  =  .666  +  ,  or  .666?;. 

Reduce  to  five  decimal  places  : 

11.    f      |      12.     jr      |      13.    ||. 

Reduce  to  equivalent  decimals. 


9. 
10. 


tIt- 
If 


I     14.    AV 


15 
16 


TVS' 


17. 

18. 


"320". 


1600 

Change  to  the  decimal  form 


19.  i  of  f 

20.  ^  of  tf . 


21.  £  of  |2| 

22.  $2J  x  tIt. 


23.  101}. 

24.  1825ft 


25.  llf    |  27.  1X3^. 

26.  8.6f  !  28.  $40ft. 


29    ^of-^- 


156  DECIMALS, 


ADDITION. 

ORAL    EXERCISES. 

286.  1.  What  is  the  sum  of  -&  and  ^  ?    .6  and  .4  ? 

2.  What  is  the  sum  of  t§-q  and  -3^-  ?    .11  and  .15  ? 

3.  What  is  the  sum  of  .12  and  .20  ?    .15  and  .25  ? 

4.  Find  the  sum  of  6  mills  and  9  mills.     .008  and  .021. 

5.  What  is  the  sum  of  .4  and  .09  ?     Of  .04  and  .009  ? 

How  many  decimal  figures  in  the  sum  of  tenths  and  tenths  ?  Of 
tenths  and  hundredths  f  Of  hundredths  and  thousandths  ?  Of  tenths 
and  thousandths  ?  In  adding  several  decimals,  each  having  a  dif- 
ferent number  of  decimal  places,  how  many  places  will  there  be 
in  the  sum  ? 

287.  Since  decimals  and  integers  increase  and  decrease 
uniformly  by  the  scale  of  10,  decimals  expressing  like 
parts  of  a  unit  may  be  added,  subtracted,  multiplied,  and 
divided  in  the  same  manner  as  integers. 

The  pupil  should  obtain  and  express  all  results  in  decimal  form. 

WRITTEN    EXERCISES. 

288.  1.  Find  the  sum  of  12.07,  326.2086,  .768,  and  1.9. 

operation.  Analysis. — Write  the  numbers  so  that  units 

12.0700  °f  the  same  order  stand  in  the  same  column. 

o  o  a   9  0  8  fi  After  reducing  the  decimals  to  a  common  de- 


7680 


nominator  by  annexing  ciphers  (280,  5),  or 
supposing  them  to  be  annexed,  add  as  in  inte- 
1.9000         gerSj  placing  the  decimal  point  before  tenths  in 
340.9466  lV     thesum. 
In  like  manner  find  the  sum 

2.  Of  .375,  .24,  .536,  .0437,  .50039,  and  .008236. 

3.  Of  405.327,  64.03,  .84673,  121.8,  and  7.00327. 

4.  Of  $18.19,  $142,095,  $.964,  $5,125,  and  $40.50. 


ADDITION.  157 

Rule. — I.  Write  the  numbers  so  that  units  of  the  same 
order  stand  in  the  same  column  and  the  decimal  points  in 
the  same  vertical  line. 

II.  Add  as  in  addition  of  integers,  and  place  the  deci- 
mal point  before  the  order  of  tenths  in  the  sum. 

5.  What  is  the  sum  of  37  thousandths,  54  ten-thou- 
sandths, 407  hundred-thousandths,  and  12345  millionths? 

6.  Find  the  sum  of  45  units,  25  tenths,  360  hundredths, 
75  thousandths,  52  ten-thousandths,  and  406  millionths. 

Find  the  sum 

7.  Of  $25f,  $81.09,  $16$,  $.87$,  $150$,  and  $$. 

8.  Of  103.60$,  6.0$,  .37012,  and  40.0034$. 

9.  Of  24.6$,  47.32^,  5.3784J,  and  2.6487SJ. 

10.  Of  61.843  acres,  8^  acres,  21.04  acres,  15$f  acres; 
and  3 1  acres. 

11.  Bought  a  ton  of  coal  for  $7f,  a  barrel  of  sugar  for 
I28TV,  a  chest  of  tea  for  $23.08,  and  a  barrel  of  flour  for 
§10.87$.     What  was  the  cost  of  all  ? 

In  the  reduction  of  each  fraction,  carry  the  decimal  to  at  least 
five  places,  to  insure  accuracy  in  the  fourth. 

lk.  Find  the  sum  of  -£-$,  f,  $f,  -fa,  and  ^-,  in  deci- 
mals, correct  to  the  fourth  place. 

13.  A  man  bought  a  farm  for  $6736.75,  which  was 
1325-f  less  than  he  sold  it  for.     What  did  he  sell  it  for  ? 

14.  How  many  rods  of  fence  will  enclose  a  field,  the 
sides  of  which  are  respectively  34.72  rods/  48$$  rods, 
152.17  rods,  95$  rods,  and  56$  rods  ? 

15.  Paid  for  building  a  house  $3450.75,  for  painting 
the  same  $518$,  for  furniture  $1204.37$,  and  for  carpets 
$810$.     What  was  the  cost  of  the  whole  ? 


158  DECIMALS 


SUBTEACTIOE". 

ORJLL    exercises. 

289.  1.  From  ■&  take  -&.    From  .9  take  .7. 

2.  From  -fifo  take  ^V.     From  .36  take  .12. 

3.  From  Tfjhj-  take  y^.     From  .028  take  .010. 

4.  From  -ffc  take  yV     From  45  cents  take  20  cents. 

5.  Find  the  difference  between  |  and  T60-.     £  and  .25. 

6.  Find  the  value  of  TV  —  .3  ;  of  .5  —  J  ;  .(35  —  .5. 

7.  Find  the  value  of  $|  —  30  cents  j  80  cents  —  $.6. 

How  many  decimal  places  in  the  remainder,  if  there  are  three  in 
the  minuend  and  one  in  the  subtrahend  ?  If  two  in  the  minuend 
and.  four  in  the  subtrahend?  If  none  in  the  minuend  and  three  in 
the  subtrahend  ? 

WRITTEN     EXERCISES, 

290.  1.  From  3.16-subtract  .2453. 

OPEBATION.  Analysis. — Write  the  given  numbers  as  in  Addi- 

3.1600  tion,  the  subtrahend  under  the  minuend,  reducing 

^  a  *  q  the  decimals  to  a  common  denominator,  by  annexing 

— - ciphers  (280,  5),  or  supposing  them  to  be  annexed, 

2.9147  and  then  subtract  as  in  integers. 

2.  From  324.07  take  70.20681.     * 

3.  From  $1034  take  $500.94. 

Rule. — I.  Write  the  subtrahend  under  the  minuend,  so 
that  units  of  the  same  order  stand  in  the  same  column. 

II.  Subtmct  as  in  subtraction  of  integers,  and.  place  the 
decimal  point  before  the  orders  of  tenths  in  the  remainder. 

Find  the  difference,  decimally,  between 

4.  $16}  and  $43yV  6.  $143-J  and  $304.96.      . 


5.  1.0066  and  .630482. 


7.  2  and  .00345. 


MULTIPLICATION.  159 


14.  .93^  and  1.169|. 

15.  1 J  and  1875  millionths. 

16.  8200  and  $70ff. 

17.  .4  and  .04J. 

18.  ft  and  ^ 

19.  .1-^  and  .Olf . 


8.  10.0402  and  26  millionths. 

9.  115  and  115  tenths. 

10.  5  and  125  ten-millionths. 

11.  $.875  and  $£. 

12.  $Hand$.75. 

13.  7.005  and  .7005. 

20.  A  speculator  having  7346  acres  of  land,  sold  at  dif- 
ferent times  364£  acres,  1235.125  acres,  2700|  acres,  and 
850.65  acres.     How  much  had  he  left  ? 

21.  A  man  bought  an  overcoat  for  $36f ,  a  sack  for  $18f , 
,    and  pants  for  $8.12 J,  and  gave  in  payment  one  fifty,  and 

two  ten-dollar  bills.     What  change  should  he  receive? 
Find  the  decimal  value 

22.  Of  $350  —  $38J  +  $100^- 

23.  Of2i--H+(.9-A). 

24.  Of  .37|  +  i  +  4.2  -  (2  -  .68}). 

25.  Of  $250  -  ($170^V  -  $14i)  +  Hi 

26.  Of  $48^  +  $.97  -  ($£  +  $.62£  +  $£). 


1/ 


MULTIPLICATION. 

ORAL      EXERCISES. 

291.  1   What  is  5  times  ^  ?    6  times .  3  ?    4  times  .5  ? 

2.  What  is  7  times  ^  ?    5  times  .08  ?    6  times  .09  ? 

3.  What  is  ^  x  3  ?    3x.7?    4x.6?    .5x7? 

4.  What  is  yf^  x  5  ?     5  x  .04?     .05  x  7  ?    g  x  .06  ? 

5.  What  is  ^x  A?    .4x.3?    .8x.7?    .6x.9? 

6.  WhatiSjIo-xA?    .5x.05?    .12x.6?    .7x.ll? 

7.  Whatis^xrk?    .03x.07?    .15x.06? 

8.  What  is  8  times  $.6  ?     7  times  yfg-  of  a  dollar  ? 

9.  What  is  8  x  .5  ?     8  x  .05  ?    8  x  .005  ?     8  x  .0005  ? 


160  DECIMALS. 

How  many  decimal  places  in  the  product  of  units  multiplied  by 
tenths?  Tenths  by  tenths?  Tenths  by  hundredths?  Hundredths 
by  hundredths  ? 

If  there  are  two  decimal  figures  in  the  multiplicand,  and  tm  in 
the  multiplier,  how  many  are  there  in  the  product?  If  threein  the 
multiplicand  and  one  in  the  multiplier?  How  many  decimal  places 
are  there  always  in  the  product  ? 

292.  Principle.— The  number  of  decimal  places  in 
any  product  is  equal  to  the  decimal  places  in  both  factors. 

WRITTEN     EXERCISES. 

293.  1.  Multiply  .64  by  .8. 

operation.  Analysis. — Multiply   as   in   fractions.      (232.) 

#64  Thus,  .64x.8  =  -,$Wt  =  tV&  =  .512.    Or, 

£  Multiply  as  in  integers,  and  since  hundredths  mul- 

— ' —  tiplied  by  tenths  produces  thousandths,  the  product 

.512  must  contain  three  decimal  places.    (Prin.) 

Multiply  Multiply 

2.  1.245  by. 27.  4.     7.25  by  .00012. 

3.  .4056  by  35.05.  5.     $506£  by  .048f 

Rule. — Multiply  as  in  multiplication  of  integers,  and 
from  the  right  of  the  product  point  off  as  many  figures  for 
decimals  as  there  are  decimal  places  in  both  factors. 

1.  If  there  are  not  as  many  figures  in  the  product  as  there  are 
decimals  in  both  factors,  supply  the  deficiency  by  prefixing  ciphers. 

2.  To  multiply  by  10,  100,  1000,  etc.,  remove  the  decimal  point  in 
the  multiplicand  as  many  places  toward  the  right  as  there  are  ciphers 
in  the  multiplier.    (266,  4.) 

Multiply  and  express  the  product  decimally  : 


6.  $324)-  by  .324. 

7.  $175.64  by  .205. 

8.  5.728  by  100. 

9.  .6207  by  1000. 


10.  5$  hundredths  by  25. 

11.  26000  by  26  thousandths. 

12.  84  tenths  by  244  hundredths. 

13.  7£  tenths  by  .06f 


MULTIPLICATION.  161 


Find  the  value 

14.  Of  3.126  x  .046  x  .3. 

15.  Of  9f  x  .07£  x  10. 

16.  Of  18.75  x  1.001  xf 

17.  Of  .25  of  -&X-04J. 


18.  Of  32?}x.9x4J. 

19.  Of  $8.56  x  .06J  x  100. 

20.  Of  18^  x  .0062^  x  1000. 

21.  Of  .01  of  i  x  100  x  .08f 


22.  Bought  156  pounds  of  cheese  at  $.12-|-  a  pound,  327 
pounds  of  coffee  at  $.26£  a  pound,  and  17  barrels  of  apples 
at  $2.87-}-  a  barrel.     What  was  the  cost  of  the  whole  ? 

23.  If  an  acre  of  land  produce  127.25  bushels  of  pota- 
toes, how  many  bushels  will  4.375  acres  produce  ? 

What  is  the  value 

24.  Of  170  barrels  of  apples,  at  $2f  a  barrel? 

25.  Of  100  cords  of  wood,  at  $4.38  a  cord  ? 

26.  Of  204^  acres  of  land,  at  $72 J  an  acre? 

27.  Of  580 J-  pounds  of  sugar,  at  9 J  cents  a  pound? 

28.  Of  126  mules,  at  $97|  each  ? 

29.  What  is  the  cost  of  3|  bales  of  cloth,  each  bale  con- 
taining 36.75  yards,  at  $.85  a  yard  ? 

30.  A  farmer  sold  300  bushels  of  oats  at  $.45  a  oushel, 
16-f  cords  of  wood  at  $3 -J  a  cord.  He  received  in  payment 
125  pounds  of  sugar  at  $.12£  a  pound,  36  pounds  of  tea 
at  $-|  a  pound,  6  barrels  of  flour  at  $8.3  7 £  a  barrel,  and 
the  remainder  in  cash.     How  much  cash  did  he  receive  ? 

Complete  the  following  equations  : 

31.  $450.75  -  $24|  x  3.24  +  $18^  =  ? 

32.  ($200  -  $125J)  x  (f  +  2.5)  =  ? 

33.  3.0065  x  .304  +  40^-  x  10  =  ? 

34.  .00493  x  1000  x  (1  -  }  +  -025)  =  ? 

35.  (4  -  .00036  +  .316)  —  (.75  +  3}  -  If)  =  ? 

36.  (rift-  x  .08J  +  .03685  x  £)  x  100  =  ? 


162  DECIMALS. 


DIVISION. 

OMA.Tj     exercises. 

294.  1.  What  is  |  of  A?    iof^fo?    iofT£fo-? 

2.  What  is  J  of  .8  ?    -J-  of  .42  ?    \  of  .072  ? 

3.  Divide  .8  by  4  ;  .56  by  7  ;  .120  by  10  ;  .0048  by  12. 

4.  Divided  by  A.    -^  by  ^ 

5.  Divide  4.8  by  6. 

Analysis. — 4.8  equals  48  tenths,  and  \  of  48  tenths  is  8  tenth*,  or  .8. 

6.  Divide  .48  by  6  ;  .48  by  .06  ;  .048  by  .006. 

7.  Divide  ^  by  ^fc  (.6  -  .12)  j  7.5  by  2.5. 

8.  Multiply  A  by  ^  (.8  x  .9).     Divide  .72  by  .9. 

9.  Multiply  T$o  by  ^  (.08  x  .09).     Divide  .0072  by  .09. 

10.  The  product  of  two  factors  is  .096,  one  of  which  is 
.8  ;  what  is  the  other  ? 

How  many  decimal  places  in  the  quotient  when  tenths  are  divided 
by  units  f  Tenths  by  tenths  ?  Hundredths  by  tenths  ?  Thousandths 
by  hundredths  ? 

If  there  are  two  decimal  figures  in  the  divisor  and  three  in  the 
dividend,  how  many  are  there  in  the  quotient  ?  If  three  in  the  di- 
visor and  three  in  the  dividend  ?  If  none  in  the  divisor  and  three  in 
the  dividend  ?    If  two  in  the  divisor  and  none  in  the  dividend  ? 

295.  Principles. — 1.  The  dividend  must  contain  at 
least  as  many  decimal  places  as  the  divisor  before  division 
is  possible. 

2.  Since  the  dividend  is  the  product  of  the  divisor  and 
quotient,  it  contains  as  many  decimal  places  as  both  divisor 
and  quotient.     Hence, 

3.  The  quotient  must  contain  as  many  decimal  places  as 
the  number  of  decimal  places  in  the  dividend  exceeds  those 
in  the  divisor. 


division.  1 63 

written    exercises. 

296.  1.  Divide  .952  by  .7. 
operation.         Analysis. — Divide  as  in  fractions.  (238.)   Thus, 
.7). 952      .952^.7=^V%-T7TF=f¥^x¥=iM=l-36^   Or, 

Divide  as  in  integers,  and  since  the  dividend  con- 
tains  three  decimal  places,  and  the  divisor  one,  the 
quotient  must  have  two  decimal  places.    (Prin.  3.) 


Divide 

2.  81.6  by  3.6. 

3.  675  by  .15. 

4.  .952  by  4.76. 


Divide 

5.  $41.25  by  33. 

6.  $518.70  by  $14.25. 

7.  345.15  by  .075. 


Rule. — Divide  as  in  division  of  integers,  and  from  the 
right  of  the  quotient  point  off  as  many  figures  as  the  deci- 
mal places  in  the  dividend  exceed  those  in  the  divisor. 

1.  If  the  number  of  figures  in  the  quotient  be  less  than  the  excess 
of  the  decimal  places  in  the  dividend  over  those  in  the  divisor,  the 
deficiency  must  be  supplied  by  prefixing  ciphers. 

2.  If  there  be  a  remainder  after  dividing  the  dividend,  annex 
ciphers,  and  continue  the  division  :  the  ciphers  annexed  are  deci- 
mals of  the  dividend. 

'  3.  In  most  business  transactions,  the  division  is  considered  suf- 
ficiently exact  when  the  quotient  is  carried  to  4  decimal  places, 
unless  great  accuracy  is  required. 

4.  To  divide  by  10,  100,  1000,  etc.,  remove  the  decimal  point  in 
the  dividend  as  many  places  to  the  left  as  there  are  ciphers  in  the 
divisor.     (266,  3.) 

8.  Divide  88.476  by  1.2  ;  by  3.6  ;  by  .01| ;  by  1.04. 

9.  Divide  $56.05  by  .59  ;  $408.37£  by  27. 

10.  Divide  $6.45  by  $.45  ;  $52  by  $.65  ;  293.75  by  45 J. 

11.  Divide  .0026  by  .003  ;  3  by  .450  ;  75  by  1000. 

12/ What  is  the  quotient  of  75.15208  divided  by  24? 
by  .24?    by  .024?    by  .0024?    by  .00024? 

13.  Divide  $3875  by  10  ;  by  100  ;  by  1000  ;  by  10000. 


164 


DECIMALS. 


What  is  the  value  of 


14.  645.5  -f- 1000. 

15.  $1000-^1.02. 

16.  $56-^.007. 

17.  1.904-T-4.76. 


18.  3-^18|.  22.  $27-r-37£ 

19.  4.2-S-31J.        23.  .001-^100. 

20.  17-^1000.     24.  lOO-j-.OOl. 

21.  .73f-f-100.      25.  $48f^$f. 

26.  Divide  .24  by  72  ;  f  of  .24  by  &  of  .042. 

27.  If  64  tons  of  iron  cost  $4816,  how  many  tons  can 
be  bought  for  $1730.75  ? 

28.  How  many  coats  can  be  made  from  32.4  yards  of 
cloth,  allowing  2.7  yards  for  each  coat? 

29.  At  $287f  each,  how  many  horses  can  be  bought  for 
$4885.80? 

30.  If  125  bushels  of  potatoes  cost  $82 J,  how  many  bar- 
rels, each  containing  2£  bushels,  can  be  bought  for  $224.40? 

31.  If  3J  cords  of  wood  cost  $11.37J,  what  will  20£ 
cords  cost? 

32.  How  much  sugar  can  be  bought  for  $46.75,  if  |  of 
a  hundred  pounds  cost  $6|-  ? 

33.  Gave  10 j-  cords  of  wood,  worth  $4|  a  cord,  for  7.74 
barrels  of  flour.    What  was  the  flour  worth  a  barrel  ? 

34.  A  man  sold  a  horse  for  $125,  and  received  in  pay- 
ment 12 J-  yards  of  cloth  at  $3 J  a  yard,  and  the  balance  in 
tea  at  $.62f.    How  many  pounds  of  tea  did  he  receive  ? 

Find  the  second  member  in  each  of  the  following  equa- 
tions : 

35.  Of  (1.008  -f-  18  +  63  -f-  4000  x  100)  —  $  =  ? 
~7714  -r-  (.34  -  .034  x  .25  of  6)  =  ? 


36.  Of  714 

37.  Of  f48  -^  800  x  10000  +  6.4  -h  .08)  -~  .125  =  ? 

38.  Of  (34  x  .193  +  2.7  x  .4-^(4.81  —  f  of  1.662)  =  ? 

39.  Of  ($262.90  -r-  $.56)  x  .0084  +  .02£  x  100  =  ? 

40.  Of  ($1260  x  3.49)  ~-  $10.47  —  $850  4-  $6.80  =  ? 


CIRCULATING     DECIMALS.  165 

CIRCULATING    DECIMALS. 

ORAL      EXERCISES. 

297.  1.  What  are  the  prime  factors  of  10  ?     Of  100? 

2.  Change  to  the  decimal  form  i  ;  f  ;  i  ;  | ;  tV  (285.) 

What  are  tlie  prime  factors  of  each  of  the  denominators  of  those 
fractions  ? 
Are  they  the  same  as  the  prime  factors  of  10  ? 
Can  these  fractions  be  reduced  to  perfect  decimals  ? 

3.  Change  to  the  decimal  form,  extending  to  four 
places,  i;  |;  i;  A- 

Can  these  fractions  be  reduced  to  perfect  decimals  ? 
What  are  the  prime  factors  of  their  denominators  ? 

4.  Change  to  the  decimal  form,  extending  to  throe 
places,  i ;  A ;  A- 

Can  these  fractions  be  reduced  to  perfect  decimals  ? 

What  are  the  prime  factors  of  their  denominators  ? 

How  do  the  decimals  produced  by  these  fractions  differ  from  the 
decimals  produced  by  the  fractions  in  examples  2  and  3  ? 

What  kind  of  decimals  are  all  fractions  equivalent  to,  that  in 
their  lowest  terms  have  denominators  containing  the  factors  2  or  5  ? 

5.  What  figure  is  constantly  repeated  in  reducing  to  a 
decimal^?    |?     £?     &? 

G.  If  a  decimal  consists  of  3  repeated  indefinitely,  what 
fraction  is  it  equal  to  ? 

7.  Is  there  any  difference  between  J  and  f  ?  $  and  £  ? 
fand  if?    ffandfJM? 

8.  Is  there  any  difference  between  -fo  and  ffi?     f| 

andfM? 

9.  If  the  numerator  is  4444,  what  must  be  its  denom- 
inator so  that  the  fraction  may  equal  f  ? 

To  change  a  repeating  decimal  number  to  an  exact  fraction,  what 
figures  must  be  used  in  the  denominator  ? 


166  DECIMALS. 

DEFINITIONS    AND    PRINCIPLES. 

298.  A  Finite  Decimal  is  a  perfect  decimal,  or 
one  that  terminates  with  the  figures  written  ;  as,  .25,  .375. 

299.  A  Circulating  Decimal  is  a  decimal  in 
which  a  figure,  or  set  of  figures,  is  constantly  repeated  in 
the  same  order  ;  as  .333  +  ,  .727272  +  . 

300.  A  Repetend  is  the  figure  or  set  of  figures,  con- 
tinually repeated. 

The  repetend  is  written  but  once,  and  when  it  consists  of  a  single 
figure  a  point  is  placed  over  it ;  when  it  consists  of  more  than  one 
figure,  points  are  placed  over  the  first,  and  over  the  last  figure.  Thus, 
the  circulating  decimal  .666  + ,  and  .297297  + ,  are  written  .6,  and  .297. 

301.  A  Pure  Circulating  Decimal  is  a  deci- 
mal which  commences  with  a  repetend  ;  as  .7,  or  .279. 

302.  A  Mixed  Circidating  Decimal  is  a  deci- 
mal in  which  the  repetend  is  preceded  by  one  or  more  deci- 
mal places  called  the  finite  part  of  the  decimal ;  as,  .27,  or 
.04648,  in  which  .2  or  .04  is  called  the  finite  part. 

303.  The  law  for  the  formation  of  repetends  will  be 
apparent  from  the  following  : 


1.  i  -.1111+      =.1. 

5.    $    =.4444+         =.4. 

2.   -fa  -.01010+        =.0t 

6.   ff  -.2323+         -.23. 

3.  TjT=. 001001+    =.6oi. 

7.  Ui  =-135135+     -.i35. 

4.  T  J^= .00010001  +  =  .oooi 

8-  Wtt=-17281728+  =1728 

304.  Principles. — 1.  Every  fraction  in  its  lowest 
terms,  whose  denominator  contains  no  other  prime  factors 
than  2  or  5  is  equivalent  to  a  finite  decimal. 

2.  Every  fraction  in  its  lowest  terms,  whose  denomina- 
tor contains  other  prime  factors  than  2  or  5  is  equivalent 
to  a  circulating  decimal. 


CIRCULATING     DECIMALS.  167 

3.  Every  fraction  in  its  lowest  terms,  whose  denomina- 
tor contains  2  or  5  with  other  prime  factors  is  equivalent 
to  a  mixed  circulating  decimal. 

4.  Every  pure  circulating  decimal  is  equal  to  a  fraction 
whose  numerator  is  the  repetend,  and  whose  denominator 
consists  of  as  many  9's  as  there  are  places  in  the  repetend. 

WRITTEN      EXERCISES. 

305.  To  change  a  fraction  to  a  finite  or  to  a 
circulating  decimal. 

1.  Change  -fe  to  a  finite  decimal.     (385.) 

2.  Change  to  finite  decimals,  $ ;  -J  ;  ^  ;  |f  ;  fj ;  -^ ; 
and  ^VV     (Pki^.  1.) 

3.  Change  to  a  pure  circulating  decimal  ¥\. 
Operation.— ^V  =  7.000000  -h  27  =  .259259  +  =  .259.    (Prin.  2.) 

4.  Change  to  pure  circulating  decimals,  -f  j  i  '■>  A  j  if  J 

5.  Change  to  a  mixed  circulating  decimal  f. 
Operation.—!  =  50°0° +■ 6  =  .8333  +  =  .83.    (Prin.  3.) 

6.  Change  to  mixed  circulating  decimals  ^  ;  ^  ;  £| ; 
W ;  and  ^ 

7.  Change  to  finite,  or  to  circulating  decimals  the  fol- 
lowing fractions  :   A;    -I;    ^;   f£ ;   tf;   A;   ^ ;  £ . 

and  Hi- 

306.  To  change  a  pure  circulating  decimal  to  a 
Traction. 

1.  Change  .216  to  a  fraction. 
operation.  Analysis.— Since  .00i  =  ^  (303),  216 

A.  a  „ig  -   _8_      is  e(lual  to  Hii  which  reduced  to  its  lowest 

*  i  o  —  tH  —  iV      terms  equals  '&.    Hence  .'216  =  ^. 


168 


DECIMALS. 


Change  to  fractions, 

2.  .45.  4.     .297. 

3.  .66.  5.     .675. 


6.  .324. 

7.  .4158. 


Rule. —  Write  the  figures  of  the  repetendfor  the  numera- 
tor of  a  fraction,  and  as  many  9's  as  there  are  places  in  the 
repetendfor  the  denominator,  and  reduce  to  its  lowest  terms. 

In  like  manner  change  to  fractions, 
8.     .279.  10.     .6435.  12.     .95121. 

%     -32i.  11.     .i067.  13.     .923076. 

14.  Reduce    2.297  to  an  improper  fraction. 

15.  Reduce  12.081  to  an  improper  fraction. 

307.  To  change  a  mixed  circulating  decimal  to 
a  fraction. 


1.  Change  .227  to  a  fraction. 


1st. 
Or  2d. 
Or  3d. 


OPERATION. 


Analysis. — Since 
the  repetend  is  not 


but 


22  7  given  decimal. 

2  finite  part. 
.225       m  =  A 


225 

990 


_5_ 

22 


Tfsnr> 


write  the  finite 


part  and  the  repe- 
tend each  as    frac- 
tions and  add  them, 
the       reasons      for 
which    will    appear 
more  clearly  in  the 
second  solution. 
Or,  by  an  abbreviated  method  of  reducing  the  fractions  to  a  com- 
mon denominator,  2  x  99  =  2  x  100  —  2 ;  hence,  2  x  100  +  27  —  2 
=  225  is  the  numerator  of  the  equivalent  common  fraction. 

2.  Change  to  fractions,  .57  ;    .048  ;    .1004  ;    .6472. 

3.  Change  to  mixed  numbers,  7.543 ;    2.564  ;    7.0126. 


SHOUT     METHODS.  169 

Eule. — Reduce  the  finite  part  and  the  repetend  of  the 
given  decimal  each  to  the  form  of  a  fraction.  Then  add 
them,  and  reduce  to  lowest  terms.     Or, 

From  the  given  decimal  subtract  the  finite  part  for  a 
numerator,  and  for  a  denominator  ivrite  as  many  9's  as 
there  are  figures  in  the  repetend,  with  as  many  ciphers 
annexed  as  there  are  figures  in  the  finite  part.    . 

Change  to  fractions, 


4. 

.04648. 

6. 

.9285714. 

8. 

.0126. 

5. 

.7852. 

7. 

.35135. 

9. 

5.27. 

To  add,  subtract,  multiply,  or  divide  circulating  decimals,  reduce 
them  to  fractions,  and  then  perform  the  required  operation. 

For  a  fuller  development  of  "  Circulating  Decimals  "  and  "  Con- 
tinued Fractions,"  see  "  Robinson's  Higher  Arithmetic." 

SHORT  METHODS. 

OKjLL   exercises. 

308.  1.  What  part  of  $1  are  8£  cents  ?  16|  cents  ? 
12-j-  cents  ?    25  cents  ?     50  cents  ? 

2.  At  25  cents  a  pound,  what  cost  22  pounds  of  coffee  ? 

Analysis.— Since  25  cents  are  $-J,  22  pounds  will  cost  22  times 
$\,  or  $-2T3-,  equal  to  $5£,  or  $5.50.     Or, 

At  $1  a  pound,  22  pounds  will  cost  $22,  and  at  $£  a  pound,  \  of 
$22,  which  is  $5|,  or  $5.50. 

3.  What  is  the  cost  of  80  pounds  of  beef  at  12-J-  cents 
a  pound  ?    At  16f  cents  ?     Afc  20  cents  ?    At  25  cents  ? 

4.  At  33^  cents  a  can,  what  will  be  the  cost  of  25  cans 
of  sweet  corn  ?     Of  37  cans  ?     Of  54  cans  ?     Of  60  cans  ? 

5.  What  is  the  cost  of  160  pounds  of  sugar  at  6 J  cents 
a  pound  ?    At  8-J-  cents  ?    10  cents  ?    12£  cents  ? 


170  DECIMALS. 

6.  How  many  pounds  of  raisins,  at  16f  cents  a  pound, 
can  be  bought  for  $5  ? 

Analysis. — Since  16|  cents  are  $J,  $5  will  buy  as  many  pounds 
of  raisins  as  $-£r  is  contained  times  in  $5,  which  are  30  times. 
Hence,  etc. 

7.  At  $.50  a  bushel,  how  many  bushels  of  oats  can  be 
bought  for  $15  ?    For  $16|  ?    For  $25  ? 

8.  At  12J  cents  a  yard,  how  many  yards  of  calico  can 
I  buy  for  27  pounds  of  butter,  at  33-J  cents  a  pound  ? 

9.  What  is  the  cost  of  40  pairs  of  shoes,  at  $1.25  a  pair  ? 

Analysis.— At  $1  a  pair,  the  cost  would  be  $40  ;  but  since  the 
price  is  $1  +  $£,  the  whole  cost  is  $40  +  $  of  $40,  or  $50. 

10.  At  $1.50  each,  what  is  the  cost  of  48  chairs  ? 

11.  What  is  the  cost  of  60  yards  of  cloth,  at  $1.12£  a 
yard?    At$1.16|?    At  $1.25  ?    At$1.33£?     At  $2.50  ? 

12.  At  $2.25  a  pair,  what  is  the  cost  of  12  pairs  of 
shoes  ?    Of  16  pairs  ?    Of  18  pairs  ?    20  pairs  ?    25  pairs  ? 

DEFINITIONS. 

309.  Quantity,  in  commercial  transactions,  is  the 
amount  of  anything  bought  or  sold,  and  is  estimated  by 
the  number  of  times  it  contains  the  measuring  unit 

310.  JPrice  is  the  value  in  money  of  each  measuring 
unit  of  any  commodity. 

311.  Cost  is  the  value  of  the  entire  quantity. 

312.  An  Aliquot  Bart   or    JEven  JPart,  of  a 

number  is  such  a  part  as  will  exactly  divide  that  number. 
Thus,  2,  2£,  3£,  and  5,  are  aliquot  parts  of  10. 

An  aliquot  part  may  be  either  an  integer  or  a  mixed  number, 
while  a  component  factor  must  be  an  integer. 


SHORT     METHODS.  171 


Aliquot  Parts  of  One  Dollar. 


5  cents  =  gV  of  $1 
10  cents  =  T\  of  $1 
20  cents  =  \  of  $1 
25  cents  =  £  of  $1 
50  cents  =  £  of  $1 


6£  cents  =  ■&  of  $1 

8i  cents  =  ■&  of  $1 

12|-  cents  =  J  of  $1 

lGf  cents  -=  f  of  $1 

331  cents  =  £  of  $1 


WRITTEN     EXERCISES,     , 

313.  To  find  the  cost  of  a  quantity  when  the 
price  is  an  aliquot  part  of  one  dollar. 

1.  What  cost  951  bushels  of  oats,  at  $.33 J  a  bushel  ? 

operation.  Analysis.— At  $1  a  bushel,  the  cost  would  be  $951 ; 

o  \  q  k  i  but  since  the  price  is  a  of  $1  a  bushel,  the  cost  is  £ 

'- of  $951,  which  is  $317.     Or,  the  cost  is  \  as  many 

317  dollars  as  there  are  bushels,  and  ^±—%il.  Hence,  etc. 

2.  What  cost  750  slates,  at  33-J  cents  each  ?  At  25  cents  ? 

3.  At  8.50  each,  what  cost  631  shad  ?    1250  ?     1605  ? 

Bule. — Take  such  a  fractional  'part  of  the  given  num.* 
her  or  quantity  as  the  price  is  of  one  dollar. 

4.  What  is  the  cost  of  12  sacks  of  coffee,  each  sack  con- 
taining 43  pounds,  at  33  J  cents  a  pound  ? 

5.  A  merchant  sold  5  pieces  of  prints,  each  containing 
28  yards,  at  16|  cents  per  yard,  6  pieces  of  sheeting,  each 
containing  34  yards,  at  8-J-  cents  per  yard,  and  received  in 
payment  41  bushels  of  oats  at  $.50  a  bushel,  and  the  bal- 
ance in  money.    How  much  money  did  he  receive  ? 


172  DECIMALS. 

6.  At  $1.12£  a  foot,  what  cost  324  feet  of  wire  fence? 

OPERATION. 

o  \  o  o  4  Analysis. — At  $1  a  foot,  the  cost  would  be  $324  ; 

but  since  the  cost  is  $1  +  $£-,   the  entire  cost  is 

J  °'5       $324  +  I  of  $324,  which  is  $364.50. 

$364.5 

7.  At  $1.33£  each,  what  will  642  steel  shovels  cost? 

8.  What  cost  320  cloth  caps,  at  $1.20  each? 

314.  To  find  the  quantity  when  the  cost  is  given, 
and  the  price  is  an  aliquot  part  of  one  dollar. 

1.  How  many  barrels,  at  $.50  each,  can  be  bought  for 
$213? 

operation.  Analysis. — Since  $|  will  pay  for 

$213  —  $1  =  426      *  barre1,  $313  wm  pay  for  as  many 
*    ^  ^  barrels  as  $|  is  contained  times  in 

Or,      213x2=426  $213,  or  426  barrels.    Or,  since  $1 

will  pay  for  2  barrels,  $213  will  pay 

for  213  times  2,  or  42C  barrels. 

2.  How  many  baskets  of  pears  can  be  bought  for  $318, 
at  $.  334  each  ?    At  $.  50  each  ? 

3.  How  many  pine-apples  can  be  bought  for  $240,  at 
16f  cents  each  ?    At  20  cents  ?    At  25  cents  ? 

Rule. — Divide  the  cost  hy  stick  a  fraction  as  will  express 
the  price  as  an  aliquot  part  of  one  dollar. 

4.  How  many  pounds  of  cheese  can  be  bought  for  $350, 
at  6£  cents  a  pound  ?    At  8£  ?     10  cents  ?    12-J  cents  ? 

5.  How  many  cocoa-nuts,  at  $.25,  can  be  bought  for 
$150.75? 


SHORT     METHODS.  173 

315.  To  find  the  cost  when  the  quantity  and  the 
price  of  100,  or  lOOO  are  given. 

1.  What  cost  564  cedar  posts,  at  $12.25  for  100  posts? 
1st  operation.  Analysis.— At  $12.25  a  post,  the  cost 

$12.25  would  be  $12.25  x  564  =  $6909.    But 

5  q  ^  since  $12.25  is  the  price  of  100  posts, 

$6909  is  100  times  the  cost.     Hence  di- 

1  0  0)6909.00  vide  by  100  (296,  Note  4),  and  the 

$69.09  result  is  $69.09.    Or, 

Since  564  is  5  and  64  hundredths 

2d  operation.  (5  64)j  if  ^  hundred  cost  $13#25, 5.64  will 

$12.25  x  5.64  =  $69.09  cost  5.64  times  $12.25,  or  $69.09. 

If  the  price  is  by  the  thousand,  divide  the  product  by  1000,  or  re-  - 
duce  the  quantity  to  thousands  and  decimals  of  a  thousand  before 
multiplying.  , 

2.  What  is  the  cost  of  1684  pounds  of  beef,  at  $9.37J  a 
hundred  pounds  ? 

3.  What  cost  22840  railroad  ties,  at  $174.55  a  thousand  ? 

4.  How  much  is  the  freight  on  4575  pounds  of  merchan- 
dise from  New  York  to  Baltimore,  at  $.98  for  100  pounds  ? 

Rule. — Multiply  the  price  by  the  quantity  reduced  to 
hundreds  and  decimals  of  a  hundred,  or  to  thousands  and 
decimals  of  a  thousand,  and  point  off  in  the  product  as  in 
multiplication  of  decimals. 

In  business  transactions,  the  letter  C  is  sometimes  used  for  hurin 
dreds,  and  M  for  thousands,  when  the  price  is  by  the  100,  or  1000. 

What  is  the  cost, 

5.  Of  536720  bricks,  at  $8.75  per  M.  ? 

6.  Of  2108  feet  of  pine  boards,  at  $3.12£  per  C? 

7.  Of  2700  pine-apples,  at  $16}  per  100  ? 

8.  Of  875  feet  of  scantling,  at  $10|-  per  M.? 

9.  Of  2160  oysters,  at  $1.86  per  100  ? 


174  DECIMALS. 

10.  Of  3080  fence  pickets,  at  $5}  per  1000  ? 

11.  Of  28642  feet  of  timber,  at  $llf  per  M.  ? 

12.  Of  1480  pounds  of  maple  sugar,  at  $12.37£  per  100  ? 

13.  What  is  the  value  of  3700  cedar  rails,  at  $5f  per  C.  ? 

14.  What  is  the  value  of  12500  shingles,  at  $6 1  per  M.  ? 

15.  Find  the  cost  of  527  feet  of  boards  at  $15£  per  M  , 
and  of  972  feet  of  siding  at  $1.62£  per  C. 

316,  To  find  the  cost,  when  the  quantity  and 
the  price  of  a  ton  of  2000  pounds  are  given. 

1.  What  is  the  cost  of  a  load  of -hay,  weighing  2280 
pounds,  at  $18.50  a  ton  ? 

OPERATION. 

2)18.50  Analysis.— Since  $18.r)0  is  the  cost  of  2000. 

^q   05  y  of  $18.50,  or  $9.25  is  the  cost  of  1000  pounds  ; 

and  2280  pounds  will  cost  2.280  times  $9.25,  or 
%-%8         $21.09. 
$21.09 

2.  At  $4.75  a  ton,  what  will  a  load  of  plaster  weighing 
2806  pounds  cost  ? 

3.  What  is  the  freight  on  21672  pounds  of  iron,  at 
$2.80  a  ton? 

Kule. — Multiply  one-half  the  price  of  a  ton  by  the  num- 
ber of  thousands  and  decimals  of  a  thousand  in  the  given 
quantity,  as  in  315. 

4.  What  is  the  value  of  150  sacks  of  guano,  each  sack 
containing  1 62£  pounds,  at  $51|  a  ton  ? 

5.  Find  the  value  of  6340  pounds  of  Lehigh  coal,  at 
$7£  a  ton,  and  5080  pounds  of  soft  coal  at  $6J  a  ton. 

6.  At  $26.44  a  ton,  what  will  be  the  cost  of  1526 
pounds  of  bone  dust  ? 


LEDGER     ACCOUNTS 


175 


LEDGER   ACCOUNTS. 

317.  A  Ledger  is  the  principal  booh  of  accounts 
kept  by  business  men.  Into  it  are  transferred,  in  a  con- 
densed form,  all  the  items  of  the  Journal,  or  Day  Book, 
for  convenient  reference  and  preservation. 

318.  The  debits  (marked  Dr.)  are  placed  on  the  left, 
and  the  credits  (marked  Cr.)  are  placed  on  the  right. 

319.  The  Balance  of  an  Account  is  the  differ- 
ence between  the  debit  and  credit  sides.  When  this  is 
settled,  or  paid,  the  account  is  said  to  be  balanced. 

320.  Find  the  balance  of  the  following  Ledger  Ac- 
counts : 


(i-) 


(2.) 


Dr. 

Cr. 

Dr. 

Cr. 

$506.76 

$42.17 

$2371.67 

$4763.84 

194.32 

36.24 

571.84 

7061.39 

173.26 

8.42 

90.50 

8242.76 

71.32 

10.71 

2037.69 

364.96 

39.46 

94.30 

94.46 

410.31 

152.60 

347.16 

876.54 

5724.27 

71.78 

40.00 

679.81 

6317.66 

320.00 

12.94 

4930.71 

2431.27 

48.50 

271.19 

104.13 

163.55 

63.41 

500.50 

1987.67 

7063.21 

56.00 

11.44 

142.84 

451.09 

410.10 

81.92 

522.71 

200.00 

72.22 

10.10 

3114.60 

1807.36 

137.89 

107.09 

152.91 

768.72 

276.44 

207.16 

9328.42 

3024.27 

176 


DECIMALS 


ACCOUNTS    AND    BILLS. 

321.  An  Account,  in  commercial  transactions,  is  a 
record  of  debits  and  credits. 

322.  A  Debtor  is  a  person  who  owes  another 
money,  goods,  or  services. 

323.  A  Creditor  is  a  person  to  whom  money,  goods, 
or  services  are  due  from  another. 

324.  A  Bill  is  a  written  statement  of  money  paid, 

of  goods  sold  or  delivered,  or  of  services  rendered.    It  is 

sometimes  called  an  Invoice. 

An  account  or  bill  should  always  state  the  place  and  the  time  of 
each  transaction,  the  names  of  both  the  parties,  the  price  or  value 
of  each  item,  and  the  entire  cost. 

325.  A  Bill  is  receipted  when  the  words  "Re- 
ceived Payment  "  are  written  at  the  bottom,  and  the 
creditor's  name  is  signed  either  by  himself,  or  by  some 
authorized  person. 

326.  The  following  abbreviations  are  in  general  use  : 


@ 

At. 

Disc't 

Discount. 

Net    Without  disc't. 

%  or 

Acc't    Account. 

Do. 

The  same. 

No. 

Number. 

Am't 

Amount. 

Doz. 

Dozen. 

Pay't 

Payment. 

Bal. 

Balance. 

Dr. 

Debtor. 

Pd. 

Paid. 

Bbl. 

Barrel. 

Exch. 

Exchange. 

Per 

By. 

Bo't 

Bought. 

Fol. 

Folio. 

Prem. 

Premium. 

B.  L. 

Bill  of  Lading. 

Fwd. 

Forward. 

Prox. 

Next  month. 

% 

Per  cent. 

Fr't 

Freight. 

Rec'd 

Received. 

Co. 

Company. 

Ins. 

Insurance. 

Sund's 

Sundries. 

Cr. 

Creditor. 

Inst. 

This  month. 

Ult. 

Last  month, 

Com 

Commission. 

Int. 

Interest. 

Yd. 

Yard. 

Dft. 

Draft. 

Mdse. 

Merchandise. 

Yr. 

Year. 

The  character  @  is  always  followed  by  the  price  of  a  unit.  Thus, 
5  yd.  of  cloth  @  $3.25,  signifies,  5  yards  of  cloth  at  $3.25  a  yard; 
\  lb.  of  tea  @  $.90,  signifies  \  a  pound  of  tea  at  $.90  per  pound. 


ACCOUNTS     AND    BILLS 


17? 


32H,  Required   the  footings  and  balances  of  the  fol- 
lowing bills  and  accounts  : 

(1.) 

New  York,  May  10,  1875. 

A.  S.  Mann  &  Co., 

Bought  of  Halsted,  Haynes  &  Co. 

336  yd.  Muslin, @  26^    .  .  $ 

98*  "     Canton  Flannel,      .     .  "   18^    .  . 

162    "     Victoria  Gingham,      .  "   16^  .  . 

110    "     Cassimere,     ....  "   $2.87£  . 


Find  the  footing  of  this  bill. 


(2.) 


$ 


Boston,  June  20,  1876. 


Messrs.  C.  P.  Mead  &  Son, 

BoH  of  Belknap,  Bro. 


216  pairs  Boys'  Kip  Boots,      . 

1G0     "  "     Brogans,     .    .     .     . 

75     "       Women's  Fox'd  Gaiters,  . 

110     *'  "         Enameled  Boots 

6  cases  Men's  Calf  Boots,     .     .     . 

1  case    Drill,  648  yd.,      .... 

36  gross  Silk  Buttons,  ..... 


Received  Payment, 


.  @    $2.25 
.    "    *  1.25 


75.50 
.14| 

.87^ 


Belknap,  Bro. 


Mr.  Chas.  Elliott, 


(3.) 

Charleston,  S.  C,  Oct.  4,  1874. 

Bo't  of  Wm.  J.  AlKIN. 


8  bales,  ea.  485  lb.,  Ordinary  Tex.  Cotton,  @.  18^ 
6  u  "  506  "  Upland,  Middlings,  .  "  21# 
3  hhd.,  215 gal.,  N.  O.  Molasses (N. Crop),  "   60^ 


Bec'd  Payment  by  draft  on  N.  T, 

Wm.  J.  Axkin. 


178 


DECIMALS. 


Messrs.  Cook  &  Cheney, 


Chicago,  Sept.  10,  1876. 


BoH  of  Baker  &  Ellis. 


275  bbl.  Flour,  State  Superfine,  .    .     @,  $7.10 

! 

146   "         "       Minnesota  Ex.,    .     ,      «'      7.87| 

| 

94   "         "      Wisconsin  XX,.     .      *      8.12£ 

650  bu.  Wheat,  No.  1,  Red  Winter,      "      1.75 

400   "         M       Illinois,  No.  1,    .     .     "      1.82 

368   "    Com,    Southern  White,     .     "       .87| 
Rec'd  Paym't  by  note  at  4  mo., 

~* 

Baker  &  Ellis. 

(5.) 

San  Francisco,  Jan.  1,  1875. 

Mr.  James  Wilde, 

To  Hodge  and  Son,  Dr. 

1874 

i 

Sept. 

10 

« ( 

To  75  lb.  Sugar, @  12^ 

"  1  caddy  Japan  Tea,  22  lb.,     "   d8f 

Oct. 

16 

"  1  sack  Rio  Coffee,  116  lb.,      "  21.^ 

n 

« < 

"  1     "    Rice,  751b.,  .     .    ..    "     9f 
M  1  box  P.  &  G.  Soap,  60  lb.,     "   10^ 

Nov. 

1 

M  25  lb.  Mackerel,    ....     "     8^ 

<< 
<< 

tt 

15 

"  9  gal.  Molasses,    ....     "  62^ 
"  18  lb.  Soda  Crackers,    .    .     "     9f 

Dec. 

20 

"  12  "  Dried  Beef      ..."   12^ 

u 

26 

"  1  box  S.  G.  Starch,  28  lb.,      "   lOfJ* 
Rec'd  Pay't, 

$ 

Hodge  &  Son, 

Per  Hi 

NBT  SOW 

rr. 

ACCOUNTS    AND    BILLS 

(6.) 


179 


Mr.  Jacob  E.  Kent, 


Detroit,  May  28,  1877. 
To  George  W.  Parker,  Dr. 


Jan. 

G 

For  Building  Out-house  as  per  contract, 

$150 

" 

•« 

44    Extra  Labor, 

14 

50 

Mch. 

20 

'•    15  days'  work  of  self,   @.  $3£- 
"    7       "        «■    of  son,      "      1.50 

" 

" 

44    784  ft.  Boards,      .    .     "      2£  per  C. 

April 

16 

"    2  days'  work,  ..."      3.50 

it 

<  t 

"    Nails,  Hinges,  and  Sundries,    .     . 

4 

75 

$ 

Statement  of  Account. 

St.  Louis,  Nov.  6,  1875. 
Messrs.  Wood  &  Cole, 


To  Phelps  &  Dodge,  Dr. 


April 

15 

'•* 

a 

June 

21 

Aug. 

10 

Oct. 

3 

n 

" 

May 

25 

July 

14 

Sept. 

5 

12 

To  30  tons  Eng.  Iron,      .     .     @,  $34.30 

44   12  cwt.  Eng.  Blister  Steel,  "  15.25 

"   6  doz.  Hoes  (Trowel  Steel),  "  9.78 

"   30  Buckeye  Plows,    .     .     "  10.45 

"   12  Cross-cut  Saws,     .    .     "  12.12| 

44   37  cwt.  Bar  Lead,      .    .     "  6.90 


Cr. 

By  22  M.  feet  of  Boards, 
44   36  M.       "      Plank, 
"   45  M.  Shingles,      .    . 
44   Draft  on  New  York, 
44  46  C.  feet  Scantling, 


@,  $27.60 
44     18.87| 

3.62i 


1.38 


Bat.  due  Phelps  &  Dodge, 


$500 


180 


DECIMALS, 


(8.) 
Account  Current;  Balanced  by  Note. 

Geo.  B.  Damon  &  Co., 

In  %  with  Gray  &  Banks, 
Dr.  Cr. 


1876 

Aug. 

2 

Sept. 

IT 

" 

■24 

Oct. 

4 

it 

18 

" 

31 

Dec. 

15 

To  796  lb.  Butter®  $.28 
»  972  "  Cheese"  .09 
"  4S1^"  Lard  "  .12 
"  509% "  Tallow"  .16 
"  81  doz.  Eggs  "  .26 
"  15  bbl.  Salt  »«  2.40 
"  9631b.  Hams    "    .14 


j  1876 
Nov. 

3 

34 

Dec. 

1 

as 

1877 

Jan. 

2 

- 

- 

By  27  bbl.  Pears  @  $9.25 
•k  56  "  Apples  «  1.87 
M  10%  bu.  Corn  "  .70 
"  31^  M  Peas   w     1.95 

"  Note  at  3  mo.  to  Bal. 


Philadelphia,  Jan.  2,  1877. 


Gray  &  Banks. 


REVIEW. 

WRITTEN     EXAMPLES. 

328.  What  is  the  cost/ 

r.  Of  7£  barrels  of  flour,  if  4J  barrels  cost  $38 1 

2.  Of  9±  tons  of  coal,  if  .875  of  a  ton  cost  $5,635  ? 

3.  Of  14.25  yards  of  cloth,  if  36.48  yards  cost  $54.72  2 

4.  Of  100  pounds  of  pork,  if  .93  cwt.  cost  $6,975  ? 

5.  Of  25.42  acres  of  land,  if  .125  of  an  acre  cost  $15-J  ? 

6.  Of  1  ton  of  plaster,  if  1680  pounds  cost  $2,856  ? 

7.  Of  .8  of  a  pound  of  tea,  if  1  pound  cost  $.62£  ? 

8.  Of  18640  feet  of  timber,  at  $6£  per  C.  ? 

9.  Of  1375  pounds  of  potash,  at  $121£  a  ton  ? 

10.  Of  19600  bricks,  at  $9|  per  M.  ? 

11.  Of  .625  of  a  ton  of  coal,  at  $7£  a  ton  ? 

12.  Of  35  yards  of  cloth,  if  29  yards  cost  $101  J? 

13.  Of  1  bushel  of  potatoes,  if  28.8  bushels  cost  $9.60? 


REVIEW.  •  181 

14.  If  36  boxes  of  raisins,  each  containing  36  pounds, 
cost  $194.40,  what  is  the  price  per  pound  ? 

15.  What  will  be  the  freight  on  10860  pounds  of  mer- 
chandise from  New  York  to  St.  Louis,  at  $1.62£  per  0.? 

16.  How  much  must  be  paid  for  1220  feet  of  boards,  at 
$25|-  per  M. ;  1866  feet  of  scantling  at  $2.12|  per  O. ;  and 
1)525  feet  of  lath  at  $3£  per  M.  ? 

17.  If  I  pay  $1.37  a  bushel  for  wheat,  $.95  for  rye,  and 
$.  73  a  bushel  for  corn,  how  much,  of  each  an  equal  num- 
ber of  bushels,  can  I  purchase  for  $70.15  ? 

18.  Bought  27|  barrels  of  sugar  for  $453.75,  and  sold 
it  at  a  profit  of  $4.62|  a  barrel.  At  what  price  was  it  sold  ? 

19.  Three  persons  bought  645  tons  of  coal,  and  divided 
so  that  the  first  had  .375  of  it,  the  second  ■£■%,  and  the 
third  the  remainder.     How  much  did  the  third  receive  ? 

20.  What  is  814^  x  26-J-|  correct  to  5  decimal  places  ? 

21.  A  person  having  $55.92,  wished  to  purchase  an 
equal  number  of  pounds  of  tea,  coffee,  and  sugar  ;  the 
tea  at  $.87£,  the  coffee  at  $.18£,  and  the  sugar  at  $.10£. 
How  many  pounds  of  each  could  he  buy? 

22.  A  dealer  bought  240000  feet  of  lumber  at  $15.90  per 
M.,  and  retailed  it  out  at  $2£  per  C.  What  was  his  whole 
gain? 

23.  Three  hundred  seventy-five  thousandths  of  a  lot  of 
dry  goods,  valued  at  $8000,  was  destroyed  by  fire.  What 
would  a  man  lose  who  owned  .15  of  the  entire  lot  ? 

24.  Bought  150  barrels  of  flour  @  $6f,  and  350  bushels 
of  wheat  ©  $1.44.  Having  sold  105  barrels  of  the  flour 
@  $8J-,  and  all  the  wheat  at  $2.06,  at  what  price  per  bar- 
rel must  the  remainder  of  the  flour  be  sold,  to  gain  $3 63  J 
on  the  whole  investment  ? 


182  DECIMALS. 

25.  Sold  20900  feet  of  timber  for  $339.62$,  and  gained 
thereby  $78.37$.    What  did  it  cost  per  C.  ? 

26.  Reduce  M  -r-  ~j  x  f  of  $  to  a  decimal. 

27.  A  farmer  exchanged  28$  bushels  of  oats  worth  $.75 
per  bushel,  and  453  pounds  of  middlings  worth  $1$  per 
hundred,  for  12520  pounds  of  plaster.  What  was  the 
plaster  worth  per  ton  ? 

28.  A  merchant  tailor  bought  27  pieces  of  broadcloth, 
each  piece  containing  19$  yards,  at  $4.31$  a  yard ;  and 
sold  it  so  as  to  gain  $381.87$,  after  deducting  $9.62$  for 
freight.     For  what  was  the  cloth  sold  per  yard  ? 

29.  If  10$  cords  of  wood  cost  $34.12$,  what  cost  60$  cords? 

30.  If  1$  hundred  pounds  of  sugar  cost  $12f,  how 
many  pounds  can  be  bought  for  $93$,  at  the  same  rate  ? 

>      31.  Paid  $108  for  grain,  -&  of  it  being  barley  at  $.62$ 
*\[  -per  bushel,  and  f  of  it  wheat  at  $1.87$  per  bushel;  the 
rest  of  the  money  was  paid  for  oats  at  $.37^  per  bushel. 
How  many  bushels  of  grain  were  bought  ? 

32.  What  is  the  value  of  (3*i  +  ~r)  -*■  4.23  ? 

33.  A  farmer  sold  to  a  merchant  3  loads  of  hay  weigh- 
ing respectively  1826,  1478,  and  1921  pounds,  at  $17.60 
per  ton,  and  281  pounds  of  pork  at  $5.25  per  C.  He 
received  in  exchange  31  yards  of  sheeting  @  $.18,  11$ 
yards  of  cloth  @  $4.50,  and  the  balance  in  money.  How 
much  money  did  he  receive  ? 

34.  A  man  expended  $140.30  in  the  purchase  of  rye  at 
$.95  a  bushel,  wheat  at  $1.37  a  bushel,  and  corn  at  $.73 
a  bushel,  buying  the  same  quantity  of  each  kind ;  how 
many  bushels  in  all  did  he  purchase  ? 


REVIEW 


183 


35.  A  farmer  had  150  acres  of  land,  which  he  could 
have  sold  at  one  time  for  $100  an  acre,  and  thereby  have 
gained  $3900  ;  but  after  keeping  it  for  a  time  he  was 
obliged  to  sell  it  at  a  loss  of  $2250.  What  did  the  land 
cost  him  an  acre,  and  for  how  much  an  acre  did  he  sell  it  ? 

36.  Bought  2500  bushels  of  wheat  @  $1.40,  and  735 
bushels  of  oats  @  $.54  ;  I  had  1470  bushels  of  the  wheat 
floured,  and  sold  it  at  a  profit  of  $435. 87^,  and  I  sold  528 
bushels  of  the  oats  at  a  loss  of  $30.  Afterward  I  sold  the 
remainder  of  the  wheat  at  $1. 25  per  bushel^  and  of  the  oats 
at  $.45  per  bushel.    Did  I  gain  or  lose,  and  how  much  ? 


329. 


SYNOPSIS  FOE  EEVIEW. 


r  1.  Notation  and 
Numeration. 


Decimal 
Currency. 


3.  Reduction. 


A    „  ,        ( 1.  Decimal  Fractions. 

1.  Defs.     ].         tt        0. 

(2.        "       Sign. 

2.  Denominator — how  composed. 

3.  Numerator — decimal  places  in. 

4.  Two  ways  of  writing  decimals. 

5.  Value   of    decimal  figures — how 

determined. 

6.  Starting    point  in    notation   and 

numeration. 

7.  Principles,  1,  2,  3,  4,  5. 
I  8.  Rule,  I,  II. 

1.  Currency. 
jfs.     i  2.  Decimal  Currency. 
3.  Federal  Money. 
2.  271,  272,  274,  275,  277. 

^  3.  Principles,  1,  2. 

1.  Art.  279. 

2.  280,  1,  2,  3,  4,  5. 

3.  282.    283. 

L  4.  284.     285,  Rule,  I,  U. 


184 


DECIMALS. 


SYNOPSIS    FOR    REVIEW— Continued 


4.  Addition.  Rule, 

5.  Subtraction.      Rule. 


6.  Multiplication. 


7.  Division. 


8.  Circulating 
Decimals. 


2, 

3 

14 


9.  Short  Methods.  - 


I,  II. 

I,  II. 

Principle. 
Rule. 

Principles,  1,  2,  3. 
Rule. 

1.  Finite  Decimal 

2.  Circ.  Decimal. 

Definitions.',  *  ^etend. 

J  4.  Pare  Ctrc.  Dec. 

I  5.  Mixed  Circ.  Dec. 
Principles,  1,  2,  3,  4. 

306.  Rule. 

307.  Rule,  1,  2. 

f  1.  Quantity. 

Definitions.  J  J  £** 

!  3.  CW. 

t  4.  Aliquot  Part. 

Rule. 

Rule. 

Rule. 

Rule. 


2. 

313. 

3. 

314. 

4. 

315. 

1  5. 

316. 

10.    Ledger  \  1.  Definitions 

Accounts.  )  g 


1.  Ledger. 

2.  ItaZ.  0/  ^4cc0Wi£. 
Position  of  Debits  and  Credits. 


11.  Accounts  and 
Bills. 


1.  Definitions. 


.8. 


1.  Account. 

2.  Debtor. 
Creditor. 
Bui. 

Receipt  of  a  Bill. 
Mercantile  Abbreviations. 


\i 


DEFINITIONS. 

330.  A  Denominate  Number  is  a  concrete 
number,  and  may  be  either  simple  or  compound ;  as,  8 
quarts,  5  feet  10  inches,  etc. 

331.  A  Simple  Denominate  Number  consists 
of  a  unit  or  units  of  but  one  denomination  ;  as,  16  cents, 
24  hours,  30  barrels,  etc. 

333.  A    Compound   Denominate   Number 

consists  of  units  of  two  or  more  denominations  of  the 
same  nature ;  as,  10  pounds  G  ounces,  5  yards  2  feet 
8  inches,  etc. 

333.  In  integral  numbers,  and  in  decimals,  the  law 
of  increase  and  decrease  is  by  the  uniform  scale  of  10 ; 
but  in  Compound  Numbers,  the  scale  varies. 

MEASUEES. 

334.  A  Measure  is  a  standard  unit  established  by 
law  or  custom,  by  which  quantity,  such  as  extent,  dimen- 
sion, capacity,  amount,  or  value,  is  measured  or  estimated. 

Thus,  the  standard  unit  of  Measures  of  Extension  is  the  yard ; 
of  Liquid  Measure,  the  wine  gallon;  of  Dry  Measure,  the  Win- 
chester bushel;  of  Weight,  the  Troy  pound,  etc.  Hence  the  length 
of  a  piece  of  cloth  is  ascertained  by  applying  the  yard  measure ; 
the  capacity  of  a  cask,  by  the  use  of  the  gallon  measure  ;  of  a  bin, 
by  the  use  of  the  bushel  measure  ;  the  weight  of  a  body,  by  the 
pound  weight,  etc. 


186 


DENOMINATE     NUMBERS. 


335.  Measures  may  be  classified  into  six  kinds : 


1.  Extension. 

2.  Capacity. 

3.  Weight. 


4.  Time. 

5.  Angles  or  Arcs. 

6.  Money  or  Value. 


MEASTTKES    OF    EXTENSION. 

336.  Extension  is  that  which  has  one  or  more  of 
the  dimensions  length,  breadth,  and  thickness.  It  may 
be  a  line,  a  surface,  or  a  solid. 

337.  The  Standard  Unit  of  measures  of  extension, 
whether  linear,  surface,  or  solid,  is  the  yard. 

LINEAR    MEASURE. 

338.  Linear  or  Long  Measure  is  used  in  meas- 
uring lines  and  distances. 

339.  A  Line  has  only  one  dimension — length. 


1  Inch. 


2  Inches. 

I 


3  Inches. 


Table. 


12    Inches  (in.)   =  1  Foot    . 

.ft. 

3    Feet               =  1  Yard   . 

.  yd. 

5i  Yards,  or  )              ■    „ 
4  Feet          [    =  1  Rod     . 

.  rd. 

&20    Rods              -  1  Mile    . 

.  mi. 

mi.      rd.        ft.  in. 

1  =  320  =  5280  =  63360 
1  =      16£=      198 
1  =        12 


1.  The  Inch  is  generally  divided  into  halves,  quarters,  eighths, 
sixteenths,  and  sometimes  into  tenths  or  twelfths. 

2.  Civil  and  mechanical  engineers,  and  others,  use  decimal  divi- 
sions or"  the  foot  and  inch. 


extension.  187 

Other  Denominations. 

3  Barley-corns,  01  sizes =1  Inch.    Used  by  shoemakers. 

. ,    ,  .  TT     ,       ..    (  to  measure  the  height  of 

4  Inches  =1  Hand.  J     horses  at  the  shoulder.     • 

9  Inches  =1  Span.   Among  sailors,  8  spans  ==  1  fathom. 

21.888  Inches  =1  Sacred  Cubit. 

6  Feet  =1  Fathom.     Used  to  measure  depths  at  sea 

120  Fathoms  =1  Cable's  Length. 

3  Feet  =1  Pace. 

1.152|  Common  Miles  =1  Geog.Mi.    Used  to  meas.  distances  at  sea. 

3  Geographic  Miles      =1  League. 

60  Geographic,  or     )   _  (of  Latitude  on  a  Meridian,  or 

69.16  Statute  Miles  f  ~"       eg  e    (of  Longitude  on  the  Equator. 

360  Degrees  =the  Circumference  of  the  Earth. 

1.  A  Knot  is  1  geographical  or  nautical  mile,  used  to  measure  the 
-speed  of  vessels. 

2.  The  geographic  mile  is  fa  of  ^,  or  ^r^  °f tne  circumference 
of  the  earth.     It  is  a  little  more  than  1.15  common  miles. 

340.  Cloth  Measure  is  practically  out  of  use.  In 
measuring  goods  sold  by  the  yard,  the  yard  is  divided  into 
halves,  fourths,  eighths,  and  sixteenths. 

At  custom  houses,  in  estimating  duties,  the  yard  is  divided  into 
tentlis  and  hundredths. 

341.  Surveyors9  Linear  Measure  is  used  by 
land  surveyors  in  measuring  roads  and  boundaries  of  land. 


7.92  Inches  =  1  Link  .  .  .  I. 

25  Links   =  1  Rod    .  .  .  rd. 

4  Rods     =  1  Chain  .  .  ch. 

80  Chains  —  1  Mile   .  .  .  mi, 


Table. 

mi.     ch.       rd.         I.  in. 

1  =  80  =  320  =  8000  =  63360 
1  =      4  =    100  =      792 
1  =      25  =      198 
1  =    7.92 


1.  A  Gunter's  Chain  is  the  unit  of  measure,  and  is  4  rods,  or  66 
feet  long,  and  consists  of  100  links. 

2.  Engineers  commonly  use  a  chain,  or  measuring  tape,  100  feet 
long. 

3.  Measurements  are  recorded  in  chains  and  hundredths. 


188        DENOMINATE  NUMBERS. 

SURFACE  OE  SQUARE  MEASURE. 

343.  Surface  or  Square  Measure  is  used  in 
computing  areas  or  surfaces. 

343.  A  Surface  has  two  dimensions — length  and 
breadth. 


SQUARE  INCH 


344.  The  Area  of  a  surface  is 
expressed  by  the  product  of  the 
numbers  that  represent  these  two 
dimensions. 


1  inch. 


345.  A  Square  is  a  plane  figure 
bounded  by  four  equal   sides,  and 
haying  four  right  angles. 
A  Square  Inch  is  a  square  each  side  of  which  is  1  inch  in  length. 

Table. 

144    Square  Inches  (sq.  in.)  =  1  Square  Foot    ....  sq.ft. 
9    Square  Feet  =  1  Square  Yard    .     .     .     .   sq.  yd. 

30£  Square  Yards  —  1  Square  Rod  or  Perch   .  sq.  rd.,  P. 

160    Square  Rods  =  1  Acre A. 

sq.  mi.    A.        sq.  rd.        sq.  yd.  sq.  ft.  -*%q.  in. 

1  =  640  =  102400  =  3097600  =  27S78400  =  4014489600 

346.  Surveyors9  Square  Measure  is  used  b/ 
surveyors  in  computing  the  area  or  contents  of  land. 

Table. 

625  Square  Links  (sq.  1.)  =  1  Pole     ....  P. 

16  Poles  =  1  Square  Chain    .  sq.  ch. 

10  Square  Chains  =  1  Acre     .     .     .     .  A. 

640  Acres  =  1  Square  Mile      .  sq.  mi. 

36  Square  Miles  (6  miles  square)  =  1  Township    .    .  Tp. 

Tp.     sq.  mi.        A.  sq.  ch.  P.  sq.  I. 

1  =   36   =  23040   =   230400   =   3686400   =   2304000000 


EXTENSION.  189 

1.  The  Acre  is  the  unit  of  land  measure. 

2.  Measurements  of  land  are  commonly  recorded  in  square  miles, 
acres,  and  hundredtJis  of  an  acre. 

For  Notes  and  Applications,  see  "Measurements"  (467,  468). 

CUBIC    OR    SOLID    MEASURE. 

347.  Cubic  or  Solid  Measure  is  used  in  com- 
puting thp  contents  or  volume  of  solids. 

348.  A   Solid    or  Body  has   three  dimensions — 
length,  breadth,  and  thickness. 

349.  The  Volume  of  a  body  is  expressed  by  the 
product  of  the  numbers  that  represent  these  dimensions. 

350.  A  Cube  is  a  body 
bounded  by  six  equal  squares, 
called  faces. 

The  sides  of  these  squares  are 
called  the  edges  of  the  cube. 

A  Cubic  Inch  is  a  cube  each 
edge  of  which  is  1  inch  in  length. 


1  inch. 


Table. 


1723  Cubic  In.  {cu.  in.)  -  1  Cubic  Ft.   .  cu,  ft.   I  cu-  Vd-  cu-fi-   cu-  *»• 
27  Cubic  Ft.  =1  Cubic  Yd.  .  cu.  yd.  !      1    —    27  =  46G56 

351.  Wood  Measure  is  used  to  measure  wood  and 
rough  stone. 

Table. 
16    Cubic  Feet  =  1  Cord  Foot cd.  ft. 

2    ^dFrt,0r^  =  ICard Cd. 

128    Cubic  Feet       \ 

„..„*.-.  *  \  Perch  of  Stone,  )  D  , 

21J  Cubic  Feet  =  1  |  or  o£  MaS0nl7  '  (    •    •  ** 

For  Notes  and  Applications,  see  "Measurements"  (474^77). 


190  DENOMINATE     NUMBEKS. 

ORAL    EXERCISES. 

352.  1.  How  many  inches  in  3  feet  ?    In  2  ft,  6  in.  ? 

2.  How  many  feet  in  48  in.  ?    In  67  in.  ?    In  75  m.  ? 

3.  In  5  yd.,  how  many  feet  ?     In  6£  yd.  ?     In  7J  yd.  ? 

4.  How  many  quarters  in  3  yd.  2  qr.  ?  Eighths  in  5  qr.  ? 

5.  At  6  cents  a  quarter,  what  cost  3  yd.  3  qr.  of  cord  ? 

6.  How  many  yards  in  96  in.  ?    In  25  ft.  ?    1ALO8  in.  ? 

7.  In  22  yd.,  how  many  rods  ?    In  3  rd.,  how  many  ft.  ? 

8.  If  a  vessel  sail  4  leagues  an  hour,  how  many  hours 
will  she  he  in  sailing  75  miles  ? 

9.  How  high  is  a  horse  that  measures  16  hands  ? 

10.  How  many  fathoms  deep  is  a  body  of  water  that 
requires  45  ft.  of  line  to  measure  it  ? 

11.  A  vessel  sunk  in  9£  fathoms  of  water :  what  was 
the  depth  of  the  water  in  feet  ? 

12.  What  part  of  a  foot  are  9  in.  ?  Of  a  yard  are  12  in.  ? 

13.  How  many  rods  is  -J  of  a  mile  ?    J  ?    £  ?    £  ? 

14.  What  part  of  a  mile  are  80  rods  ?    32  rd.  ?    64  rd.  ? 

15.  At  $£  a  foot,  what  will  6  yd.  1  ft.  of  lead  pipe  cost  ? 

16.  What  part  of  a  mile  are  20  oh.  ?    Are  60  ch.  ? 

17.  At  $|  a  rod,  what  will  it  cost  to  dig  a  trench  J-  of  a 
mile  long  ? 

18.  How  many  square  yards  in  54  sq.  ft.  ?  In  84  sq.  ft.  ? 

19.  In  a  piece  of  zinc  12  in.  long  and  9  in.  wide,  how 
many  square  inches  ? 

20.  ^Find  the  difference  of  6  ft.  square,  and  6  sq.  ft.  ? 

21.  In  a  lot  12  rd.  long  and  10  rd.  wide,  how  many 
square  rods  ?     What  part  of  an  acre  ? 

22.  How  many  yards  of  carpeting  a  yard  wide,  will 
cover  a  floor  15  ffc.  long  and  12  ft.  wide  ? 


EXTENSION.  191 

23.  What  will  it  cost  to  pave  a  court  10  ft.  by  15  ft., 
at  $.50  a  square  foot  ? 

24.  At  20  cents  a  square  yard,  what  will  it  cost  to  paint 
a  ceiling  18  ft.  by  10  ft.  ? 

25.  How  many  cubic  feet  in  2  cu.  yd.  ?    In  3  cu.  yd.  ? 

26.  How  many  cubic  inches  in  1  cu.  ft.  20  cu.  in.  ? 

27.  What  part  of  a  cubic  yard  are  9  cu.  ft.?  Are  12  cu.  ft.? 

28.  How  many  cubic  feet  in  3  cd.  ft.  ?    In  4  cd.  ft.  ? 

29.  In  J  of  a  cord,  how  many  cord  feet?     Cubic  feet? 

30.  In  2  perch  of  stone,  how  many  cubic  feet  ? 

31.  How  many  cubic  inches  in  a  10  inch  cube  ? 

32.  What  is  the  difference  between  4  cubic  inches,  and 
a  4  inch  cube  ? 

33.  How  many  blocks,  each  containing  1  cu.  ft.,  are 
equal  to  a  block  6  ft.  long,  5  ft.  wide,  and  3  ft.  thick  ? 

MEASUEES    OF    CAPACITY. 

353.  Capacity  signifies  extent  of  room  or  space. 

354.  Measures  of  capacity  are  divided  into  two  classes  ; 
Measures  of  Liquids  and  Measures  of  Dry  Substances. 

355.  The    Units   of   Capacity  are    the    Gallon  for 
Liquid,  and  the  Bushel  for  Dry  Measure. 

LIQUID    MEASURE. 

356.  Liquid   Measure    is   used   in    measuring 
liquids. 


Table. 

4  Gills  (gi.)  =  1  Pint  .  .  .  pt. 
2  Pints  sr  1  Quart  .  .  qt. 
4  Quarts       =  1  Gallon    .     .  gal. 


gal.    qt.    pt.     gi. 

1  =  4  =  8  =  32 

1=2=8 

1  =    4 


192  DENOMINATE     NUMBERS. 

In  estimating  the  capacity  of  cisterns,  reservoirs,  etc. 

hhd.  bbl.    gal.      qt.        pt. 
31|  Gal.  =  1  Barrel    .     .    .  bbl.  1  =  2  =  63  =  252  =  504 

63    Gal.  =  1  Hogshead  .    .  hhd.  1  =  3H=  126  =  252 

1.  The  barrel  and  hogshead  are  not  fixed  measures,  but  vary 
when  used  for  commercial  purposes. 

2.  The  tierce,  hogshead,  pipe,  butt,  and  tun  are  the  names  of 
3asks,  and  do  not  express  any  fixed  measures.  They  are  usually 
gauged,  and  have  their  capacities  in  gallons  marked  on  them. 

357.  Apothecaries9  Fluid  Measure  is  used 
in  prescribing  and  in  compounding  liquid  medicines. 

Table. 

60  Minims,  or  drops  (1U)  =  1  Fluidrachm    .     .    ft.. 

8  Fluidrachms                  =  1  Fluidounce     .     .    /g . 
16  Fluidounces  ==  1  Pint 0. 

8  Pints  ==  1  Gallon    ....     Gong. 

Cong.  1  =  0.  8  =fl  128  =  /3  1024  =  HI  61440. 

1.  Gong.,  for  congius,  is  the  Latin  for  gallon  ;  0.,  for  octarius,  is 
the  Latin  for  one-eighth. 

The  minim  is  equivalent  to  a  drop  of  water.  A  pint  of  water 
weighs  a  pound. 

Drops  are  indicated  in  a  physician's  prescription  by  gtt. 

The  symbols,  as  in  Apothecaries'  Weight,  precede  the  numbers 
to  which  they  refer  ;  thus,  0.  3/  §  6,  is  3  pints  6  fluid  ounces. 

DRY    MEASURE. 

358.  Dry  Measure  is  used  in  measuring  dry  arti- 
cles, such  as  grain,  fruit,  roots,  salt,  etc. 

Table. 

bu.   pk.     qt.      pt. 

2  Pints  (pt.)  =  1  Quart    .    .   qt.  1  =  4  =  32  =  64 

8  Quarts        =  1  Peck      .     .  pk.  1  =    8  =  16 

4  Pecks  -  1  Bushel  .    .  bu.  1=2 

For  Notes  and  Applications,  see  "  Measurements."    (482.) 


CAPACITY.  193 


ORAL     EXERCISE 


359.  1.  How  many  gills  in  3  pints  ?     In  2  qt.  1  pt.  ? 

2.  How  many  pints  in  1  gal.  ?     In  1  gal.  2  qt.  1  pt.  ? 

3.  In  36  pints,  how  many  quarts  ?    How  many  gallons  ? 

4.  What  part  of  a  quart  are  6  gi.  ?..  What  part  of  a  gallon  ?  ^. 

5.  What  part  of  2  gal.  are  4  pints  ?jjAxe  8  pt.  M  qt  ?  ! 

6.  How  many  gills  in  £  of  a  quart  ?-_ln  -J  of  a  gallon  ?  ^V 

7.  How  many  pints  in  64  gills ?    How  many  quarts? 
Gallons  ?J>  •/■* 

8.  How  many  fluidrachms  in  5  fluidounces?      )h® 

9.  How  many  pint  bottles  will  be  required  to  hold  3  gal. 

1  qt.  of  syrup  ?     2  gal.  3  qt.  ?  Xl>  -  1  -2- 

10.  At  5  cents  a  pint,  what  will  2  gal.  of  milk  cost  ? 

11.  If  10  gal.  2  qt.  are  drawn  from  a  barrel  of  vinegar, 
how  many  gallons  remain  ?  &* 

12.  If  a  gallon  of  wine  cost  $6,  what  will  3  pt.  cost  ?  'J,  1y 

13.  How  many  barrels  can  be  filled  from  20  hogsheads  ?  g^ 

14.  At  20  cents  a  quart,  how  many  gallons  of  molasses 
will  U  buy ?<T $6?^  $5.60?    ']  v 

15.  How  many  pints  in  6  quarts  ?     In  2  pk.  1  qt.  ?> 

16.  How  many  quarts  in  3  pk.  6  qt.  ?'y0In  1  bu.  2  pk.  ?  ^ 

17.  In  96  qt.,  how  many  pecks  ?l^How  many  bushels?    3 

18.  What  part  of  5  bu.  are  5  pk.  ?  W)f  1  bu.  are  12  qt.  ? 

19.  How  many  quart  boxes  will  1  bu.  2  pk.  6  qt.  fill  ?    ^j 

20.  At  20  cents  a  quart,  what  will  J-  bu.  of  plums  cost  ?   ,^ 

21.  At  5  cts.  a  pt.,  what  is  a  bushel  of  chestnuts  worth :    a  ^ 

22.  At  $3.20  a  bushel,  how  many  quarts  of  peanuts  can 
be  bought  for  $2  ? 

23.  Bought  \  bu.  of  chestnuts  for  $1£,  and  sold  them 
for  8  cents  a  pint.    What  was  the  gain  ? 

9 


194  DENOMINATE     NUMBERS. 

v 

MEASUEES    OF    WEIGHT. 

360.  Weight,  on  the  earth,  is  the  measure  of  gravity, 
and  varies  according  to  the  quantity  of  matter  a  body 
contains. 

361.  The  Standard  Unit  of  weight  is  the  Troy 
pound  of  the  Mint,  and  contains  5760  grains. 

TROY    WEIGHT. 

362.  Troy  Weight  is  used  in  weighing  gold,  silver, 
and  jewels,  and  in  philosophical  experiments. 

Table. 

lb.  oz.  pwt.  gr. 
24  Grains  (gr.)  =  1  Pennyweight,  pwt.  1  =  12  =  240  =  5760 
20  Pennyweights  =  1  Ounce    .    .    .  oz.  1  =    20  =    480 

12  Ounces  =  1  Pound  .    .    .  lb.  1  =      24 

A  Carat  is  a  weight  of  about  3.2  Troy  grains,  and  is  used  to  weigh 
diamonds  and  precious  stones. 

The  term  carat  is  also  used  to  express  the  fineness  of  gold,  and 
means  a  twenty-fourth  part.  Thus,  gold  is  said  to  be  18  carats  fine, 
when  it  contains  18  parts  of  pure  gold,  and  6  parts  of  alloy,  or  baser 
metal. 

.  APOTHECARIES'    WEIGHT. 

363.  Apothecaries9  Weight  is  used  by  physicians 
and  apothecaries  in  prescribing  and  mixing  dry  medicines. 

Table. 

20  Grains  (gr.  xx)   =  1  Scruple    .    .    .    .  «c,  or   9, 

3  Scruples  Oiij)  =  1  Dram dr., -or    3. 

8  Drams  (  3  viij)    =  1  Ounce oz.,  or    §. 

12  Ounces  ( I  xij)  =1  Pound..  .  .  .  lb.,  or  tb. 
Ibl  =  1 12  =  3  96  =  3288  =  gr.  5760. 

1.  Medicines  are  bought  and  sold  by  Avoirdupois  Weight. 

2.  The  pound,  ounce,  and  grain  are  the  same  as  those  of  Troy 
Weight,  the  ounce  being  differently  divided. 


WEIGHT 


195 


3.  Physicians  write  prescriptions  according  to  the  Roman  nota- 
tion, using  small  letters,  preceded  by  the  symbols,  writing  j  for  i, 
when  it  terminates  a  number.  Thus,  6  ounces  is  written,  §  vj  : 
8  dr.,  3  viij  ;  14  sc,  Bxiv.,  etc. 

4.  IJ  is  an  abbreviation  for  recipe,  or  take  ;  a,  aa. ,  for  equal  quan- 
tities ;  ij.  for  2  ;  ss.  for  semi,  or  half  ;  gr.  for  grain  ;  P.  for  particula, 
or  little  part ;  P.  seq.  for  equal  parts  ;  q.  p.,  as  much  as  you  please. 


AVOIRDUPOIS    WEIGHT. 
364.  Avoirdupois  Weight  is  used  for  weighing 
all  coarse  and  heavy  articles. 

Table. 


T.  cwt.    lb.        oz. 
1=20=2000=32000 
1=  100=  1600 
1=      16 


16  Ounces  (oz.)         =1  Pound  .    ...  lb. 

100  Pounds  =1  Hundred- weight  cwt 

20  Cwt.,  or  2000  lb.=l  Ton T. 

1.  The  Ounce  is  often  divided  into  halves,  quarters,  etc. 

2.  The  long,  or  gross  ton,  hundred-weight,  and  quarter  were  for- 
merly in  common  use  ;  but  they  are  now  seldom  used,  except  in 
estimating  duties  at  the  U.  S.  Custom  Houses,  and  in  weighing  a 
few  of  the  coarser  articles,  such  as  coal  at  the  mines,  etc. 

Long  Ton  Table. 

T.  cwt.  qr.      lb.        oz. 


16  Ounces  =  1  Pound  .  lb. 

28  Pounds  =  1  Quarter  .  qr. 

4  Quarters  =  1  Hund.  .  cwt. 

20  Cwt.,  or  2240  lb.  =  1  Ton    .  .  T. 


1=20=80=2240=35840 

1=  4=  .112=  1792 

1=    28=     448 

1=      16 


3.  Both  custom  and  the  law  of  most  of  the  States  make  100  pounds 
a  hundred-weight. 

365.  The  following  denominations  are  also  in  use  : 

100  Pounds  of  Grain  or  Flour  make    1  Cental. 


100 

Dry  Fish 

tt 

1  Quintal. 

100 

'         Nails 

" 

lKeg. 

196 

'         Flour 

" 

1  Barrel. 

200 

'         Pork  or  Beef 

" 

1  Barrel. 

280 

Salt  at  N.  Y.  S. 

works     ' 

1  Barrel. 

240 

'         Lime 

<t 

ICask. 

196 


DENOMINATE     NUMBERS 


366.  The  weight  of  the  bushel  of  certain  grains  and 
roots  has  been  fixed  by  statute  in  many  of  the  States  ; 
and  these  statute  weights  must  govern  in  buying  and  sell- 
ing, unless  specific  agreements  to  the  contrary  are  made. 

Table  of  Avoirdupois  Pounds  in  a  Bushel, 

As  prescribed  by  statute  in  the  several  States  named. 


COMMODITIES. 


Barley 

Beans 

Blue  Grass  Seed 

Buckwheat 

Castor  Beans 

Clover  Seed 

Dried  Apples 

Dried  Peaches... 

Flaxseed 

Hemp  Seed 

Indian  Corn 

Indian  Corn  in  ear 
Indian  Corn  Meal 

Mineral  Coal 

Oats 

Onions 

Potatoes 

Eye 

Rye  Meal 

Salt 

Timothy  Seed.... 
Wheat 


54 


56 


q  te;^:^ 


48,48  48  48 
6060:60 


60 


141414 


46,4646 
60  60  60 
24  25  34 
83  88  68 
555656 
44  44  44 
52  56  50 
70  68 
48  50 
80  70 
32:32  35 
57  48  57 
6060,60 
5456  56 


50  50 
45  45,45 

60  60  60 


GO 


50 


^  * 


3  I 


50 


46  48,48 


40 


42  42 


ill* 


48   48 


40 


60  60,60 

28  28  241 


50 


32  32 

521 


56  56 

50 


50 


66 


48 


d*9 


Ufa 


46  47 


42 


55  56 
58  56 


:;2 


60 
56 

56 

44| 
60  GO 


no 


GO 


46  42 


50 


50 


1 


&  tt  (£ 


46  45  48 


56 


50 


50 


42 


00 


2828 
28J28 

56 

56  56 


50, 
60  60 
56  56 


46 
60  63 


1.  In  Pennsylvania  80  lb.  coarse.  70  lb.  ground,  or  62  lb.  fine  salt 
make  1  bushel ;  and  in  Illinois,  59  lb.  common,  or  55  lb.  fine  salt 
make  1  bushel. 

2.  In  Maine  64  lb.  of  ruta-baga  turnips,  or  of  beets  make  1  bushel. 


WEIGHT.  197 

ORAL      EXERCISES.**) 

367.  1.  How  many  grains  in  3  pwtv?     In  3  5  ? 

2.  How  many  ounces  in  60  pwt.  ?  In  100  pwt.  ?  120  pwt.  ? 

3.  How  many  ounces  in  5  lb.  ?    In  3  lb.  10  oz.  ?  4=1  lb.  ? 

4.  How  many  ounces  in  40  drains  ?  In  04  dr.  ?  120  dr.  ?'d 

5.  How  many  pounds  in  36  oz.  F  In  70  ozr?     110  oz.  ? 

6.  How  many  scruples  in  10  drams  ? \  In  80  grains?. 

7.  What  will  a  gold  chain,  weighing  1  oz.  12  pwt.,  cost 
at  $1  a  pennyweight  ? 

8.  What  part  of  a  pound  Troy  are  4  oz.  ?  6  oz.  ?  8  oz.  ? 

9.  How  many  parts  of  pure  gold  in  a  ring  16  carats  fine  ?  /& 

10.  How  many  powders  of  8  grains  each,  can  be  made 
from  half  an  ounce  of  medicine  ? 

11.  How  many  tablespoons,  each  weighing  2  oz.,  can 
be  made  from  2  lb.  10  oz.  of  silver  ?  J  y  f/x^~ 

12.  How  many  pills  of  gr.  5  each  can  be  made  from  3  1 
3  2  of  calomel  ? 

13.  What  is  the  value  of  a  gold  bracelet  weighing  3  ozV^y  j 
15  pwt.,  at  $20  an  ounce  ?         ■  %  \&  / 

14.  How  many  ounces  in  4  lb.  Avoir.?     In  5  lb.  6  oz.  ? 

15.  How  marfy  pounds  in  7  cwt!/  ?    In  84  cwt.  ?  V  ^> 

16.  How  many  cwt.  in  600  lb.  r'ln  350  lb.  ?   In  875  lb.? 

17.  In  3  T.,  how  many  hundred-weight  ?  How  many  lb.? 

18.  What  part  of  a  cwt.  are  25  lb.  ?    50  lb.  ?     75  lb.  ? 

19.  How  many  cwt.  in  J  of  a  ton  ?     In  -J  of  a  ton  ? 

20.  How  many  tons  are  50, cwt.  ?     80  cwt.  ?     95  cwt.  ? 

21.  What  will  a  ton  of  hay  cost,  at  1  cent  a  pound  ? 

22.  At  8  cents  an  ounce,  what  will  1\  lb.  of  licorice  cost  ? 

23.  What  will  f  lb.  of  candy  cost,  at  3  cents  an  oz.?        .3 

24.  At  $2  a  bushel,  what  must  be  paid  for  2  bags  of 
wheat,  each  containing  120  lb.  ? 


198 


DENOMINATE     NUMBERS, 


368. 


SYNOPSIS    FOR    REVIEW. 


r  1.  Definitions 
I. 

2. 


{'• 


H  1 


QQ 


Denominate  Number.      2.    Simple 
Number.    3.  Comp.  Denom.  Number. 

Definition  of  Measure. 
Classification. 

1.  For  what  used.  2.  Table. 

3.  Other  Denominations. 

4.  Surveyors'  Linear  Meas. 
1.  For  what  used. 

it.  Surface. 
2.  Area. 
3.  Square. 

3.  Table. 

4.  Surveyors'  Square  Meas. 
1.  For  what  used. 

j  1.  Solid.    2.  Vol 

L  3.  Table. 

For  what  used. 
Table. 


1.  Linear  Meas- 
ure. 


2.  Square  Meas- 
ure. 


3.  Cubic  Meas- 

ure. 

4.  Wood  Meas- 

ure. 


2.  Defs. 


jl.F, 
j  2.  T; 


r  1. 

Definition  of  Capacity. 

2. 

Units  of  Capacity. 

3 

3. 

f  1.  For  what  used. 
Liquid  Meas-  J  3   Table 

o 

UBB-                1   3.  Apoth.  Fluid  Measure. 

3 

4. 

(   1.  For  what  used. 
Dry  Measure,  j  g  TaMe 

'  1. 

Definition  of  Weight. 

2. 

Standard  Unit. 

o   < 

3. 

(  1.  For  what  used. 
Troy  Weight,  j  3  Table 

4. 

Apothecaries  j  1.  For  what  used. 

1 

3 

Weight.        (  2.  Table. 

5. 

Avoirdupois     (  1.  For  what  used. 

JO 

Weight.       \  2.  Table. 

TIME 


199 


60  Seconds  (sec 
60  Minutes 

0 

24  Hours 

7  Days 
365  Days,  or 
12  Calendar  Months 

366  Days 
100  Years 

yr.    mo. 
1  =  12  = 

da. 
,365 

MEASUEES    OF    TIME. 

369.  Time  is  a  measured  portion  of  duration. 

370.  The  Unit  of  measure  is  the  mean  solar  day. 

Table. 

=  1  Minute min. 

=  1  Hour hr. 

=  1  Day da. 

=  1  Week    .....  wk. 

=  1  Common  Year    .    .  yr. 

=  1  Leap  Year  .  .  .  yr. 
=  1  Century  .  ...  (7. 
hr.         min.  sec. 

8760  =  525600  =  31536000 
.  366  =  8784  =  527040  =  31622400 
In  most  business  transactions  30  days  are  considered  a  month,  and 
12  months  a  year.     Four  weeks  are  sometimes  called  a  lunar  month. 
The  calendar  year  is  divided  as  shown  in  the  diagram  : 

1.  The  Solar  Day  is 
the  interval  of  time 
between  two  succes- 
sive passages  of  the 
sun  across  the  meri- 
dian of  any  place. 

2.  The  Mean  Solar 
Day  is  the  mean  or 
average  length  of  all 
the  solar  days  in  the 
year. 

3.  The  Civil  Day, 
used  for  business  pur- 
poses and  which  cor- 
responds with  the 
mean  solar  day,  begins 
and  ends  at  12  o'clock, 
midnight.  A.M.  de- 
notes the  time  before  noon ;  M 


365  or  366  days, 
at  noon ;  and  P.  M. 


afternoon. 


200 


DENOMINATE     NUMBERS 


4.  The  Solar  Tear  is  exactly  365  da.  5  hr.  48  min.  49.7  sec. 

5.  The  Common  Year  consists  of  365  da.  for  3  successive  years, 
every  fourth  year  containing  366  da.,  one  day  being  added  for  the 
excess  of  the  solar  year  over  365  da.  This  day  is  added  to  the  month 
of  February,  which  then  has  29  da.,  and  the  year  is  called  Leap-year. 

371.  The  following  rale  for  leap  year  will  render  the 
calendar  correct  to  within  1  day,  for  a  period  of  4000  years  : 

I.  Every  year  exactly  divisible  by  4  is  a  leap  year,  the 
centennial  years  excepted;  the  other  years  are  common  years. 

Thus,  1876  is  a  leap  year,  but  1877  is  a  common  year. 

II.  Every  centennial  year  exactly  divisible  by  400  is  a 
leap  year  ;  the  other  centennial  years  are  common  years. 

Thus,  the  year  2000  is  a  L.  year,  but  1800  and  1900  are  com.  years. 


CIRCULAR    MEASURE. 

372.  Circular  or  Angular  Measure  is  used  in 
measuring  angles  and  arcs  of  circles,  in  determining  lati- 
tude and  longitude,  the  location  of  places  and  vessels,  etc. 

373.  The  Unit  is  the  Degree,  which  is  -^  part  of 
the  circumference  of  any  circle. 

374.  A  Circle  is  a  plane 
figure  bounded  by  a  curved 
line  every  point  of  which  is 
equally  distant  from  a  point 
within  called  the  Center. 

375.  The  Circumfer- 
ence of  a  circle  is  the  line 
that  bounds  it. 

376.  An  Arc  is  any  part 
of  the  circumference;  as 
A  D,  D  E. 


MISCELLANEOUS. 


201 


377.  An  Angle  is  the  difference  in  the  direction  of 
two  lines  proceeding  from  a  common  point  called  the  ver- 
tex.   Thus,  A  C  D  and  D  C  B  Fig.  2. 

are  angles,  and  0  is  their  vertex. 

378.  A  Right  Angle  is 
formed  by  drawing  one  line  per- 
pendicular to  another.  Thus, 
ACE  and  EC  Bare  right  angles.     -5 

379.  A  Degree  is  one  of  the  360  equal  parts  into 
which  the  circumference  of  a  circle  is  supposed  to  be 
divided.  Thus,  E  and  B  (Fig.  1)  are  at  the  distance  of 
90°,  or  a  right  angle  from  each  other,  the  vertex  being  at 
the  center  of  the  circle. 

380.  The  Measure  of 
an  Angle  is  the  arc  of  the 
circle  included  between  its  sides. 
Thus,  the  arc  D  B  (Fig.  3)  is  the 
measure  of  the  angle  D  C  B. 


Fig.  3. 


Table. 


60  Seconds  (") 
60  Minutes 
30  Degrees 
12  Signs,  or  360° 


Cir.  8      ° 
1=12=360=21600=1296000 
1=  30=  1800=  108000 
1=      60=        360 
1=  60 


=  1  Minute  .     '  » 
=  1  Degree  .    ° 
=  1  Sign  .    .    8 
=  1  Circle     .     Cir. 

1.  A  Semi-Clrcum .  is  one-half  of  a  circumference,  or  180°. 

2.  A  Quadrant  is  one-fourth  of  a  circumference,  or  90°. 

3.  A  Sextant  is  one-sixth  of  a  circumference,  or  60°. 

4.  A  Sign  is  one-tioelfth  of  a  circumference,  or  30°. 

5.  A  degree  varies  with  the  size  of  the  circle  ;  thus,  a  degree  of 
long,  at  the  Equator  is  69.16  statute  miles,  at  30°  of  latitude  it  is 
59.81  mi.,  at  60°  of  latitude  it  is  34.53  mi.,  and  at  90°,  it  is  nothing. 

6.  A  minute  of  the  earth's  circumference  is  called  a  geographic, 
or  nautical  mile,  and  is  a  small  fraction  less  than  1.16  common  miles. 


202  DENOMINATE     NUMBERS. 

COUNTING. 

381.  The  following  table  is  used  in  counting  certain 
classes  of  articles : 

12  Units  or  things  =  1  Dozen    .     .  .  doz. 

12  Dozen  =  1  Gross     .     .  .  gro. 

12  Gross  =  1  Great  Gross  .  O.  gro. 

20  Units  or  things  =  1  Score      .    .  .  Se. 

.  1.  Two  things  of  a  kind  are  often  called  a  pair,  and  six  things  a 
set ;  asa  pair  of  horses,  a  set  of  chairs,  spoons,  etc. 

PAPER. 

382.  The  denominations  of  the  following  table  are  used 
in  the  paper  trade : 


24  Sheets  =  1  Quire. 

20  Quires  =  1  Ream. 

2  Reams  =  1  Bundle. 

5  Bundles  =  1  Bale. 


1  Bale  =    5  Bundles. 

1  Bundle  =    2  Reams. 

1  Ream  =  20  Quires. 

1  Quire  =  24  Sheets. 


BOOKS. 

383.  The  terms  folio,  quarto,  octavo,  etc.,  indicate  the 
number  of  leaves  into  which  a  sheet  of  paper  is  folded. 


When  a  sheet  is 

The  hook  is 

And  1  sheet  of 

folded  into 

called 

paper  makes 

2  leaves 

a  Folio, 

4  pp.  (pages). 

4      " 

a  Quarto  or  4to, 

8   " 

8      " 

an  Octavo  or  8vo, 

16   " 

12      " 

a  Duodecimo  or  12mo, 

24   " 

16      " 

a  16mo, 

32   " 

18      " 

an  18mo, 

36   " 

Clerks  and  copyists  are  usually  paid  by  the  folio    for  making 
copies  of  legal  papers,  records,  and  documents. 

72  words  make  1  folio,  or  sheet  of  common  law. 
90  1  chancery. 


MISCELLANEOUS.  203 

ORAL      EXERCISES. 

384.  1.  How  many  seconds  in  J  min.  ?    In  §  min.  ? 

2.  How  many  minutes  in  120  sec.  ?    In  180  sec.  ? 

3.  How  many  hours  in  90  min.  ?    In  200  min.  ? 

4.  How  many  hours  in  2  da.  ?    In  3|  da.  ?    In  5  J  da.  ? 

5.  How  many  hours  from  6  a.m.  to  5  p.  m.  ? 

6.  In  4  wk.  3  da.,  how  many  days?    In  5  wk.  4  da.  ? 

7.  How  many  minutes  from  10  min.  past  9  o'clock  to 
25  min.  past  10  a.  m.  ? 

8.  How  much  time  from  20  minutes  before  11  A.  m.  to 
half  past  10  o'clock  p.  m.  ? 

9.  Which  of  the  months  have  30  da.  each  ?  31  da.  each  ? 

10.  How  many  days  from  Jan.  1  to  March  5,  inclusive  ? 

11.  How  many  days  from  May  10  to  July  16,  inclusive  ? 

12.  Which  of  the  following  are  leap  years,  and  which 
are  common  years  :  1874?  1876?  1880?  1886?  1900? 

13.  How  many  centuries  and  years  since  the  birth  of 
Christ  ? 

14.  How  many  leap  years  in  every  century  ?  %.  o  <*JD  6 

15.  How  many  degrees  in  J  a  circle?  In  £  ?  In  £  ?  In  |  ? 

16.  How  many  geographic  miles  in  2P°?    in3°?   In4°?2 

17.  How  many  common  miles  in  6  geographic  miles? 

18.  How  many  degrees  in  360  nautical  miles  ?     b 

19.  How  many  degrees  in  |  a  quadrant  ?  }  1  n  £  a  sextant  ? 

20.  How  many  degrees  in  f  of  a  circumference  ?x  L  *i 

21.  What  part  of  a  circumference  are  60°  ?  90°  ?  180°  ?  *- 

22.  How  many  dozens  in  2£  gross  ?  "  In  3  J  gro.  ? 

23.  How  many  dozens  in  \  of  a  great  gross  ?  ,  In  f  ? 

24.  How  many  score  in  100  ?    Pairs  in  50  ?     Sets  in  75  ? 

25.  In  1  B.  of  paper,  how  many  reams  ?    How  many 
quires  ? 


204  DENOMINATE     NUMBERS. 

26.  How  many  eggs  in  5|  dozen  ?     In  12  doz.  and  7  ? 

27.  How  many  quires  of  paper  in  \  a  ream  ?  '  In  3 \  rm.  ?o? 

28.  How  many  years  in  4  score  years  and  10  ?  ^  ^ 

29.  How  many  sheets  of  paper  will  be  required  to  make 
a  12mo  book  of  320  pages  ?    Of  480  pages  ? 

30.  How  many  sheets  will  be  required  to  make  a  quarto     C 
book  of  144  pages  ?     Of  240  pp.  ?     Of  360  pp.  ? 

31.  How  many  16mo  books  will  the  paper  for  1  quarto  . 
book  make  ?  ^ 

MEASUKES    OF    VALUE. 

385.  Money  is  the  measure  of  the  value  of  things, 
and  is  used  as  a  medium  of  exchange  in  trade. 

38G.  Specie  or  Coin  is  metal  struck,  stamped,  or 
pressed  with  a  die,  to  give  it  a  fixed  legal  value,  and 
authorized  by  Government  to  be  used  as  money. 

387.  JPaper  Money  consists  of  bills  and  notes  duly 
authorized  by  Government  to  circulate  as  substitutes  for, 
or  representatives  of,  money. 

388.  Currency  is  a  term  applied  to  all  kinds  of 
money  employed  in  trade  and  commerce,  both  of  coin  and 
paper. 

389.  A  Mint  is  a  place  in  which  the  coin  of  a  coun- 
try or  government  is  manufactured. 

390.  An  Alloy  is  a  metal  compounded  with  another 

of  greater  value.    In  coinage,  the  less  valuable  or  baser 

metal  is  not  reckoned  of  any  value. 

Gold  and  silver,  in  a  pure  state,  are  too  soft  for  coinage  ;  hence 
they  are  hardened  by  compounding  them  with  an  alloy  of  baser 
metal,  while  their  color  and  other  valuable  qualities  are  not  im- 
paired. 


E.        %           d.           Ct.            77?. 

et. 

1  =  10  =  100  =  1000  =  10000 

d. 

1  =  10  =  100  =  1000 

$. 

1  =   10  =   100 

E. 

1  =   10 

VALUE.  205 


UNITED    STATES    MONEY. 

391.  United  States  Money  is  the  legal  currency 
of  the  United  States,  and  is  sometimes  called  Federal 
Money. 

392.  The  Unit  of  U.  S.  Money  is  the  Gold  Dollar. 


10  Mills  (m.)  =  1  Cent  , 
10  Cents  —  1  Dime  , 
10  Dimes  =  1  Dollar 
10  Dollars      =  1  Eagle  , 

Federal  money  was  adopted  by  Congress  in  1786.  Previous  to 
this,  pounds,  shillings,  and  pence  were  in  use.  There  is  no  coin  for 
the  mill. 

393.  The  Coin  of  the  United  States  consists  of  gold, 
silver,  nickel,  and  bronze,  and  is  as  follows : 

394.  Gold.  The  double-eagle,  eagle,  half-eagle, 
quarter-eagle,  three-dollar  and  one-dollar  pieces. 

395.  Silver.  The  Trade-dollar,  one-dollar,  half-dol- 
lar, quarter-dollar,  twenty-cent,  and  the  ten-cent  pieces. 

396.  Nickel*    The  five-cent,  and  three-cent  pieces. 

397.  Bronze.    The  one-cent  piece. 

1.  The  half-dime  and  three-cent  pieces,  the  bronze  two-cent,  and 
the  nickel  one-cent  pieces  are  no  longer  coined. 

2.  The  Trade-dollar  weighs  420  grains,  and  is  designed  solely  for 
purposes  of  commerce  and  not  for  currency.  The  legal-tender  dollar 
weighs  412^  grains. 

3.  The  Standard  purity  of  the  gold  and  silver  coins  is,  9  parts  (.9) 
pure  metal,  and  1  part  (.1)  alloy.  The  alloy  for  gold  coins  is  silver 
and  copper,  the  silver,  by  law,  not  to  exceed  -^  of  the  whole  alloy. 
The  alloy  of  silver  coins  is  pure  copper. 

4.  The  five  and  three  cent  pieces  consist  of  3  parts  (.75)  copper, 
and  1  part  (.25-  nickel. 

5.  The  one  cent  piece  consists  of  .95  copper,  and  .05  of  zinc  and  tin. 


206  DENOMINATE     NUMBERS. 

CANADA    MONEY. 

398.  Canada  Money  is  the  legal  currency  of  the 

Dominion  of  Canada,  and  is  a  decimal  currency. 

The  denominations  are  dollars,  cents,  and  mills,  and  have  the  same 
nominal  value  as  the  corresponding  denominations  of  U.  S.  Money. 

399.  The  Coin  of  the  Dominion  of  Canada  is  silver  and  bronze. 

400.  The  Silver  Coins  are  the  fifty-cent,  twenty-five-cent, 
ten-cent,  and  five-cent  pieces. 

401.  The  Bronze  Coin  is  the  one-cent  piece. 

1.  The  gold  coins  in  use  are  the  Sovereign  and  the  Half- Sovereign. 

2.  The  intrinsic  value  of  the  50-cent  piece  in  United  States  coin 
is  about  461  cents,  of  the  25-cent  piece  23^  cents.  In  ordinary 
business  transactions,  they  pass  the  same  as  United  States  coin. 

ENGLISH    MONEY. 

402.  English  or  Sterling  Money  is  the  legal 
currency  of  Great  Britain. 

403.  The  Unit  of  Eng.  Money  is  the  Pound  Sterling. 

Table. 


£.    s.      d.    far. 

1=20=240=960 

1=  12=  48 

1=    4 


4  Farthings  {far.)  =     1  Penny    .    .    .    .  d. 

12  Pence  =     1  Shilling ....  a. 

(1  Sovereign,  or  .     .so 

20SMtogs  =|1Poundg\    .    .£ 

The  value  of  a  Sovereign  in  United  States  money  is  $4.8665. 
The  character  for  pound  (£)  is  written  before  integers  ;  5  pounds 
=  £5. 

Other  Denominations. 

2  Shillings  (*.)      =      1  Florin  .     .     .    .    fl. 
5  Shillings  =      1  Crown  .     .    .    .     cr. 

404.  The  Coin  of  Great  Britain  in  general  use  consists  of  gold, 
silver,  and  copper,  as  follows  : 

405.  Gold.    The  sovereign  and  half-sovereign. 

406.  Silver.     The  crown,  half-crown,   florin,   shilling,   six. 
penny,  and  three-penny  piece. 

407.  Copper.     The  penny,  half-penny,  and  farthing. 


VALUE.  20? 

FRENCH    MONEY. 

408.  French  Money  is  the  legal  money  of  France 
and  is  a  decimal  currency. 

409.  The  Unit  of  French  Money  is  the  Silver  Franc. 

Table. 

fr.     dc.      ct.  m. 

10  Millimes  (m.)  =   1  Centime    .    .  ct.  1  =  10  =  100  =  1000 

10  Centimes         =    1  Decime      .     .  dc.  1  =    10  =    100 

10  Decimes  =    1  Franc    .    .     .  fr.  .  1  =      10 

The  value  of  a  franc  in  U.  S.  money  of  account  is  $.193. 

410.  The  Coin  of  France  consists  of  gold,  silver,  and  bronze. 

411.  Gold.     The  100,  40,  20,  10,  and  5  franc  pieces. 

412.  Silver.  The  5,  2,  and  1  franc,  the  50  and  the  2.r  centime 
pieces. 

413.  Bronze.    The  10,  5, 2,  and  1  centime  pieces. 

GERMAN    MONEY. 

414.  The  Empire  of  Germany  has  adopted  a 
new  and  uniform  system  of  coinage. 

415.  The  Unit  is  the  "Mark"  (Reichsmark),  equal 
to  23.85  cents,  U.  S.  Money. 

A  pound  of  gold  .900  fine  is  divided  into  139|  pieces,  and  the  ^ 
part  of  this  gold  coin  is  called  a  "  Mark,"  and  this  is  subdivided 
into  100  pennies  (Pfennige). 

416.  The  Coin  of  the  New  Empire  consists  of  gold,  silver,  and 
nickel,  and  is  as  follows  : 

417.  Gold.    The  20,  the  10,  and  the  5-mark  pieces. 

418.  Silver.    The  2,  and  the  1-mark,  and  the  20-penny  pieces. 

419.  Nickel.  The  10,  and  the  5-penny,  and  pieces  of  less 
Valuation. 

The  10-mark  piece  (gold)  is  equal  to  3^  P.  Thalers  (old). 
The  1-mark  (silver)  is  equal  to  10  S.  Groschen,  or  100  pennies. 
The  20-penny  (silver)  is  equal  to  2  S.  Groschen,  or  \  of  a  mark. 
The  10-penny  (nickel)  is  equal  to  1  S.  Groschen,  or  -fa  of  a  mark. 


208 


DENOMINATE     NUMBERS 


420. 


SYNOPSIS    FOE    REVIEW. 


a 

a 

a 

1 

« 

DO 


^     r 


p 

Si 


m 

O  ai 
H  fa 


8  W 


1.  Definition  of  Time. 

2.  Standard  Unit.      8.  Table. 
4.  Rule  for  Leap  Year,  I,  II. 

f  1.  For  what  Used. 

2.  Standard  Unit. 

(  1.  Circle.   2.  Circumference, 

3.  Defs.         -j       3.  Arc.    4.  Right  Angle. 

(       5.  Degree.  » 

4.  Measure  of  an  Angle.    5.  Table. 


1.  Counting. 
j  2.  Paper. 
3.  Books. 

1.  Defs. 

2.  U.  S. 
Monet. 

3.  Canada 
Money. 

4.  English 
Money. 

5.  French 
Money. 

6.  German 
Money. 


Table. 
Table. 
Table. 

1.  Money.  2.  Specie.  3.  Pa- 
per Money.  4.  Currency, 
5.  Mint.     6.  Alloy. 

1.  Definition. 

2.  Unit.    3.  Table. 
4.  Coin. 

1.  Definition. 

2.  Denominations. 

3.  Coin. 

1.  Definition. 

2.  Unit.     3.  Table. 

4.  Coin. 

1.  Definition. 

2.  Unit.    3.  Table. 
4.  Coin. 

1.  Definition. 

2.  Unit. 

3.  Coin. 


REDUCTION.  209 


REDUCTION. 

421.  Reduction  of  Denominate  Numbers 

is  the  process  of  changing  their  denomination  without 
altering  their  value. 

422.  Denominate  numbers  may  be  changed  from 
higher  to  lower  denominations,  or  from  lower  to  higher 
denominations. 

423.  To  reduce  denominate  numbers  from  high- 
er to  lower  denominations. 

ORAL    EXERCISES. 

How  many  inches  in  3  ft.  ?  In  5  ft.  ?  In  4  ft.  10  in.? 
How  many  feet  in  5  yd.  2  ft.  ?    In  1  rd.  10  ft.  ? 
Eeduce  12  fath.  4  ft.  to  feet.     15-J-  hands  to  inches. 
How  many  quarters  in  3-J-  yd.  ^  How  many  eighths  ? 
How  many  chains  in  1£  mi.'?    How  many  rods  ? 
In  3£  sq.  yd.,  how  many  sq.  ft.  ?  In  7  sq.  yd.  5  sq.  ft.? 
In  10  A.,  how  many  sq.  ch.  ?     How  many  sq.  rd.  ? 
Change  3  cu.  yd.  to  cu.  ft.     3  cd.  ft.  to  cubic  feet. 
Change  4  cords  to  cord  feet.     2  perch  to  cubic  feet. 
How  many  quarts  in  2  gal.  3  qt.  ?  ;  In  5£  gal.  ? 
In  4  bu.  1  pk.,  how  many  pecks  ?   Quarts  ?   Pints  ? 
In  2  pints,  how  many  fluidounces  ?    Fluidrachms  ? 
How  many  pin  ts  in  3  pk.  ?     In  2  pk.  6  qt.  ?    vuA 
How  many  half-pecks  in  1£  bu.  ?     In  3 J  bu.  ?  *  * 

In  5  lb.  Troy,  how  many  ounces  ?   In  5  lb.  Avoir.  ? 
How  many  pounds  in  5  cwt.  20  lb.  ?    In  4|  cwt.  ? 
In  5  dr.,  how  many  scruples  ?    How  many  grains  ? 
In  2  bu.  20  lb.  of  wheat,  how  many  pounds  ? 

mo 


2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

\1 

11. 

12. 
13. 

14. 

15. 

16. 

17. 

18. 

210  DENOMINATE     NUMBEBS. 

19.  How  many  minutes  in  5  hr.  40  min.  ?     In  4|  hr.  ? 

20.  In  \  a  sign,  how  many  degrees  ?  Geographic  miles  ? 

21.  In  10  gross  9  doz.,  how  many  dozen  ?    In  7-J-  gro.?  ^  ^ 

22.  In  2  reams  of  paper  how  many  quires  ?     Sheets?^  £  V 

23.  In  £5  10s.,  how  many  shillings  ?     In  3£  sov.  ? 

24.  In  5  francs  how  many  centimes  ?     In  10£  francs  ? 

25.  How  many  pence  in  \\  crowns  X\   In  12  florins  ? 

26.  How  many  crowns  in  £5  ?   CHow  many  florins  ? 

27.  How  many  pennies  in  3  marks  ?    In  5£  marks  ? 

424.  Principle. — Denominate  Numbers  are  changed 
to  lower  denominations  by  Multiplication. 

WRITTEN    EXERCISES. 

425.  1.  Reduce  28  rd.  4  yd.  2  ft.  10  in.  to  inches. 

operation.  Analysis. — Since  1  rd.  is  equal 

2  8  rd.  4  yd.  2  ft.  10  in.      to  5£  yd. ,  there  are  in  28  rd.  5|  times 


H 


as  many  yards  =  154  yd  ,  to  which 
add  4,  the  given  number  of  yards ; 
1  5  8  yd.  28  x  5£  +  4  =  158,  the  number  of 

3  yards  in  28  rd.  4  yd. 

*~n  fi.  Since  1  yd.  is  equal  to  3  ft.,  there 

are  in  158  yd.  3  times  as  many  feet 

12  =  474  ft.,  to  which  add  2,  the  given 

5  7  2  2  in.  number  of  feet;  158x3 +  2=47  6,  the 

number  of  feet  in  28  rd.  4  yd.  2  ft. 
Since  1  ft.  is  equal  to  12  in.,  there  are  in  476  ft.  12  times  as  many 
inches  =  5712.    To  this  add  10,  the  given  number  of  inches  ;  476  \  12 
+ 10  =  5722,  the  number  of  inches  in  28  rd.  4  yd.  2  ft.  10  inches. 

2.  Reduce  7  lb.  10  oz.  16  pwt.  11  gr.  to  grains. 

3.  In  3  T.  6  cwt.  21  lb.  12  oz ,  how  many  ounces? 

4.  How  many  inches  in  12  fathoms  3  ft.  10  in.? 

5.  Change  6  wk.  5  da.  9  hr.  25  min.  to  minutes. 


REDUCTI  OK. 


211 


Kule. — I.  Multiply  the  units  of  the  highest  denomina- 
tion of  the  given  number,  by  that  number  of  the  scale  that 
zuill  reduce  it  to  the  next  lower  denomination,  and  to  the 
product  add  the  number  of  that  denomination  given, 

II.  Proceed  in  like  maimer  tvith  this  and  each  successive 
denomination  obtained,  until  the  number  is  reduced  to  the 
required  denomination. 


Eeduce 

6.  12  mi.  36  rd.  10  ft.  to  ft. 

7.  10  rd.  5J-  ft.  to  inches. 

8.  27J  yd.  to  eighths. 

9.  1  A.  15  sq.  yd.  to  sq.  ft. 

10.  2  sq.  mi.  125  A.  to  P. 

11.  14  sq.  mi.  to  acres. 

12.  3  mi.  51  ch.  6  1.  to  links. 

13.  75  Cd.  6  cd.  ft.  to  cu.  ft. 

14.  12  hhd.  21  gal.  to  pt. 

15.  24  bu.  3  pk.  to  quarts. 

16.  Cong.4,  0.5,  f  ^8tof3. 

17.  31|gal.  to  gills. 


Change 

18.  7  T.  9  cwt.  18  lb.  to  lb. 

19.  22  lb.  10  oz.  to  pwt. 

20.  Jb  16,   ?7,   3  3,  to  3. 

21.  1  common  year  to  min. 

22.  The  summer  mos.  to  sec. 

23.  1  leap  year  to  hours. 

24.  10  S.  22°  5'  to  min. 

25.  5  bundles  to  quires. 

26.  6  G.  gro.  to  dozens. 

27.  326£  sov.  to  pence. 

28.  26 \  fr.  to  centimes. 

29.  £34£  to  pence. 


30.  How  much  is  5  lb.  9  oz.  14  pwt.  of  gold  dust  worth, 
at  $.75  a  pwt.? 

31.  How  many  rods  of  fence  will  enclose  a  farm  £  of  a 
mile  square  ? 

32.  If  1  barrel  will  hold  2  bu.  3  pk.,  how  many  barreU 
will  be  required  to  hold  1548  bu.  1  pk.  ? 

33.  How  many  boxes,  each  containing  12  lb.,  can  be 
filled  from  a  hogshead  containing  9  cwt.  60  lb.  of  sugar  : 

34.  If  I  buy  9  bu.  of  chestnuts  at  $4|  a  bushel,  and 
retail  #iem  at  12J  cents  a  pint,  what  is  my  whole  gain  ? 


212  DENOMINATE     NUMBERS. 

35.  How  many  times  will  a  wheel  1G£  ft.  in  circumfer- 
ence revolve  in  running  42  miles  ? 

36.  How  many  minutes  less  in  every  autumn  of  a  com- 
mon year  than  in  either  spring  or  summer  ? 

37.  If  it  require  4  reams  10  quires  of  paper  to  print  a 
book,  how  many  sheets  are  required  ? 

38.  At  12£  cents  each,  what  will  be  the  cost  of  2  great 
gross  of  writing  books  ? 

39.  If  a  clock  tick  seconds,  how  many  times  will  it  tick 
during  February,  1877  ? 

40.  If  your  age  is  21  yr.  2G  da.,  how  many  minutes  old 
are  you,  if  5  leap  years 'have  occurred  in  that  time  ? 

41.  If  a  vessel  sail  120  leagues  in  a  day,  how  many 
statute  miles  does  she  sail  ? 

42.  How  mauy  pint,  quart,  and  2-quart  bottles,  of  each 
an  equal  number,  can  be  filled  from  a  barrel  of  31£  gallons? 

.43.  In  the  eighteenth  century,  how  many  hours? 

44.  How  large  an  edition  of  a  12mo  book  can  be  printed 
from  2  bales,  2  bundles,  15  quires  of  paper,  allowing  8 
sheets  to  the  volume  ? 

45.  How  many  pages,  2  pages  to  each  leaf,  will  there 
be  in  an  8vo  book,  containing  1G  fully  printed  sheets? 

How  many  pounds 


4G.  In  36 \  centals  of  grain  ? 

47.  In424bbl.  of  flour? 

48.  In  29.5  quintals  of  fish  ? 
£9.  Inll6£bbl.  of  salt? 
50.  In  63.25  kegs  of  nails? 

What  is  the  value  in  U.  S.  Money 

56.  Of  28  sovereigns  ?  |  58.  Of  25  francs  ? 

57.  Of  £25  10s.  ?  I  59.  Of  42*  marks  ? 


51.  In  .75  of  75  bu.  of  salt  ? 

52.  In  125J  bu.  of  wheat  ? 

53.  In  |  of  21648  bu.  oats? 

54.  In  .7  of  40  bu.  corn  meal? 

55.  In  7.5  casks  of  lime  ? 


REDUCTION".  213 

426.  To  reduce  denominate  numbers  from  lower 
to  higher  denominations. 

ORAL      EXERCISES. 

1.  How  many  feet  in  108  in.  ?     How  many  yards  ? 

2.  How  many  square  yards  in  63  sq.  ft. ?     In  85  sq.  ft? 

3.  How  many  chains  in  200  1.  ?    In  425  1.  ?    In  674 1.  ? 

4.  In  81  cu.  ft.,  how  many  cu.  yd.  ?     How  many  cd.  ft.  ? 

5.  How  many  cords  in  100  cd.  ft. ?     In  256  en.  feet? 

6.  Change  120  sq.  ch.  to  A.     80  P.  to  square  chains. 

7.  In  162  in.,  how  many  hands  ?    Spans  ?    Feet  ? 

8.  In  112  pt.,  how  many  quarts?    Pecks  ?    Bushels  ? 

9.  How  many  gallons  in  46  qt.  ?     96  pt.  ?    128  gi.  ? 

10.  Eeduce  O.  160  to  Cong. ;  f  3  90  to  f  §  . 

11.  Change  96  oz.  to  Troy  pounds  ;  to  Avoir. 

12.  Eeduce  3  45  to  oz. ;    ?  75  to  pounds. 

13.  In  400  pwt.,  how  many  oz.  ?    How  many  lb.? 

14.  In  508  lb.,  how  many  cwt.  ?     In  1276  lb.  ? 

15.  In  630  lb.  of  wheat,  how  many  bushels  ? 

16.  In  140  da.  how  many  wk.  ?  Months,  of  30  da.  each  ? 

17.  Change  1200  min.  to  hours.     84  hr.  to  days. 

18.  How  many  doz.  are  240  eggs  ?     How  many  gross  ? 

19.  How  many  degrees  in  180'?    Minutes,  are  240"? 

20.  In  90  units,  how  many  score?     Sets?    Pairs? 

21.  In  120  quires  of  paper,  how  many  reams  ?   Bundles  ? 

22.  In  120d.  how  many  shillings  ?    Crowns  ?    Florins  ? 

23.  In  500  pennies,  how  many  marks  ? 

427.  Principle. — Denominate  numbers  are  changed 
to  higher  denominations  by  Division, 


214  DENOMINATE     NUMBERS. 

WRITTEN     JEXER  C  IS  ES. 

428.  1.  Change  5722  inches  to  rods. 

~^™  .  ™™  Analysts. — Since  12  in. 

OPERATION.  .     ■  KHao    . 

19U799  .  make   1   ft.,  in  5722  in. 

1 Z  )b  J  A  A  m.  there  are  M  many  f  eet  ag 

3  )  4  7  6  ft.  -f-  1  0  in.  12  in.  are  contained  times 

k  i  \  i  c  o     ^     .    o  *j.  m  5^22  in. ,  or  476  ft.  and 

54)  158  yd.  -f  2  ft.  1A  . 

i  '  J  10  in.  more. 

2  2  Since  8  ft.  make  1  yd., 

11)316  half-yd.  in  476  ft  tliere  are  158 

'- J  yd.  and  2  ft.  more. 

2  8  rd.  +  4  yd.  And  since  5}  yd.  make 

5722  in.  =:  28  rd.  4  yd.  2  ft.  10  in.     1  rd.,  in  158  yd.  there  are 

28  rd.  and  4  yd.  more. 
In  order  to  divide  by  5|,  both  dividend  and  divisor  may  be  re- 
duced to  halves  before  dividing.     In  this  case  the  remainder,  if  any, 
is  haloes,  which  may  be  reduced  to  integers. 

2.  Eeduce  157540  minutes  to  weeks. 

3.  Reduce  80820  links  to  miles. 

4.  Change  487630  pwt.  to  pounds. 

Rule. — I.  Divide  the  units  of  the  given  denomination 
faj  that  number  of  the  scale  which,  is  equal  to  a  unit  of  the 
next  higher  denomination,  and  icrite  the  remainder  as  a 
part  of  the  answer. 

II.  In  like  manner,  divide  this  and  each  successive  quo- 
tient until  reduced  to  the  denomination  required.  The 
last  quotient,  with  the  remainders  annexed,  is  the  required 
result. 

How  many 

5.  Miles  are  3168000  in.  ? 

6.  Acres  are  256800  P.  ? 

7.  Sq.  mi.  are  27878400  sq.  ft.  ? 

8.  Cu.  ft.  are  216840  cu.  in.  ? 

9.  Cords  are  38042  cu.  ft.  ? 


Reduce 

10.  30876  gills  to  hhd. 

11.  27072  qt.  to  bushels. 

12.  66742  pt.  to  barrels. 

13.  103720  pt.  to  gallons. 

14.  f  I  8106  to  Cong. 


BEDUCTIOK. 


215 


How  many 

15.  Pounds  Troy  are  85894  gr.  ? 

16.  Tons  are  51570  pounds  ? 

17.  Cwt.  are  40607  ounces  ? 

18.  Pounds  are  3000  pwt.? 

19.  Bu.  are  12060  lb.  of  wheat  ? 

20.  Bbl.  are  3038  lb.  of  flour  ? 

21.  Bu.  are  6496  lb."  of  oats  ? 

22.  Quin.  are  3172  lb.  of  fish  ? 

23.  Weeks  are  3114061  sec.  ? 

24.  Months  are  8263420  min.  ? 

25.  Degrees  are  2007200"? 

26.  Deg.  are  5270  Naut.  mi.  ? 


Reduce 

27.  120400  pens  to  gro. 

28.  2734  eggs  to  dozens. 

29.  5020  balls  to  scores. 

30.  10738  sheets  to  rm. 

31.  6048  quires  to  bun. 

32.  24684d.  to  crowns. 

33.  4076s.  to  florins. 

34.  $194.66  to  half-sov. 

35.  42346  far.  to  £. 

36.  886,85  to  francs 

37.  $225.40  to  sov. 

38.  $47.70  to  marks. 


39.  If  the  Atlantic  Cable  is  3200  mi.  in  length,  and 
cost  10  cents  a  foot,  what  was  its  entire  cost  ? 

40.  If  a  cubic  foot  of  gray  limestone  weigh  175  lb., 
what  is  the  weight  of  a  cubic  yard  ? 

41.  What  is  the  cost  of  a  load  of  oats  weighing  1860 
lb.,  at  156  a  bushel.? 

42.  In  a  storm  at  sea,  a  ship  changed  her  longitude  423 
geographic  mi.     How  many  degrees  and  minutes  ? 

43.  How  much  time  will  a  person  gain  in  40  yr.,  by 
rising  25  min.  earlier  and  retiring  20  min.  later  every  day, 
counting  9  leap  years  in  the  time  ? 

44.  What  will  a  peck  of  clover-seed  cost,  at  $.12 J  a  lb.? 

45.  What  will  a  ton  of  corn-meal  cost,  at  $1.20  a  bu.  ? 

46.  An  Illinois  farmer  sold  a  load  of  corn  weighing 
2496  lb.,  and  a  load  of  oats  weighing  1920  lb. ;  for  the 
corn  he  received  $.62  a  bushel,  and  for  the  oats  $.44  a 
bushel.    What  did  he  receive  for  both  loads  ? 


216  DENOMINATE     NUMBEES. 

REDUCTION  OF  DENOMINATE  FRACTIONS. 

429.  A  Denominate   Fraction   is  a  fraction 

whose  integral  unit  is  a  denominate  number.     Thus,  \  of 

a  week,  .  7  of  an  acre,  are  denominate  fractions. 

The  Principles,  Operations,  and  Analyses  of  the  reduction  of  de- 
nominate fractions  are  essentially  the  same  as  those  of  denominate 
integers. 

430.  To  reduce  denominate  fractions  from  higher 
to  fractions  of  lower  denominations. 

ORA.I      EXERCISES. 

1.  Eeduce  ^  of  a  gallon  to  the  fraction  of  a  pint. 

Analysis. — Since  in  1  gal.  there  are  4  qt.,  in  T\  gal.  there  are  T^ 
of  4  qt.,  or  \  qt.  ;  and  since  in  1  qt.  there  are  2  pt.,  in  £  qt.  there 
are  \  of  2  pt.,  or  |  pt.     Hence  ^  gal.  equals  \  pt. 

2.  Eeduce  fa  lb.  Troy  to  the  fraction  of  an  oz. 

3.  What  part  of  a  pint  is  ^  of  a  qt.  ?    ^  of  a  pk.  ? 

4.  What  decimal  part  of  a  day  is  .12  of  a  week  ? 
Analysis. — Since  in  1  wk.  there  are  7  days,  in  .12  wk.  there  are 

.12  of  7  da.,  or  .84  da. 

5.  What  part  of  a  peck  is  .02  of  a  bu.?  .07  bu.  ?  .25  bu.  ? 

6.  Reduce  .5  gal.  to  the  fraction  of  a  quart  ?   Of  a  pint  ? 

7.  What  part  of  an  inch  is  ^  of  a  foot  ?    ^  of  a  yard  ? 

8.  Change  .04  of  a  pound  to  the  decimal  of  an  ounce. 

WRITTEN     EXERCISES. 

431.  1.  Reduce^  of  a  bushel  to  the  fraction  of  a  pint. 
operation.  Analysis.— Same   as    for  oral 

Ti¥lra.x4  =  *pk.        iu«sti°ns,-   (43°-).    ,    v.*  « 

118                 "      ,  Multiply  successively  by  4,  8, 

TS  P^'  *  ° . =  T  ^  and  2,  the  numbers  in  the  descend- 

-|  qt.  X  2  =  4  Pk  ing  scale  required  to  reduce  bush- 

Or,  ^  X  \  X  f  X  f  =  4  Pk  els  to  pints.    (425.) 


EEDUCTIOK.  217 

2.  Keduce  T^j  of  a  rod  to  the  fraction  of  a  foot. 

3.  Change  -g-g-g-  of  an  ounce  to  the  fraction  of  a  grain. 

4.  What  part  of  a  pint  is  -^  of  a  hogshead  ? 

5.  What  part  of  a  shilling  is  .012  of  a  £? 

Eule. — Multiply  the  fraction  of  the  higher  denomina- 
tion by  the  numbers  as  factors  in  the  descending  scale  suc- 
cessively between  the  given  and  the  required  denomination. 
(425.) 

6.  What  part  of  an  ounce  is  ftfts  °f  a  pound  Avoir.  ? 

7.  Reduce  l4740  of  an  acre  to  the  fraction  of  a  sq.  rd. 

8.  Reduce  .005  of  a  bushel  to  the  decimal  of  a  pint. 

9.  How  many  yards  is  |  of  ^  of  a  rod  ? 

10.  Change  .0000625  mi.  to  the  decimal  of  a  foot. 

11.  What  part  of  an  ounce  Troy  is  f  of  £  of  2  pounds  ? 

12.  What  part  of  a  yard  is  81*18  of  a  mile  ? 

13.  What  fraction  of  a  link  is  ft  of  a  rod  ? 

14.  What  part  of  a  minute  is  .000175  of  a  day? 

15.  What  part  of  a  sq.  rd.  is  y^  of  4£  times  ft  A.  ? 

432.  To  reduce  denominate  fractions  to  integers 
of  lower  denominations. 

ORAL     EXERCISES. 

1.  How  many  hours  in  f  of  a  day  ? 

Analysis. — Since  in  1  da.  there  are  24  hr.,  in  f  of  a  day  there  are 
|  of  24  hr.,  or  16  hr.     Hence  §  da.  equals  16  hr. 

2.  How  many  minutes  in  ft  hr.  ?    In  ft  hr.  ?    In  f  hr.  ? 

3.  How  many  quarts  in  f  pk.  ?    In  J  bu.  ?    In  ft  bu.  ? 

4.  How  many  ounces  in  .5  of  a  pound  ? 

Analysis. — Since  in  1  lb.  there  are  16  oz.,  in  .5  lb.  there  are  .5  of 
16  oz.,  or  8  oz. 
10 


218 


DENOMINATE     NUMBERS. 


5.  How  much  is  .7  hr.  ?     .25  hr.?    .15  hr.  ?    .8  hr.  ? 

6.  How  many  yards  in  -fa  of  a  rod  ?    In  §  of  a  rod  ? 

7.  How  many  cwt.  in  f  of  a  ton  ?     How  many  pounds  ? 

8.  Change  to  pints  -J  gal.    J  pk.     -^  bu.     £  of  2  pk. 

9.  Change  J  of  an  acre  to  sq.  rd.     -§-  sq.  yd.  to  sq.  ft. 

10.  How  many  pecks  in  .75  bu.  ?    Quarts  in  1.25  pk.  ? 

11.  Change  j  lb.  to  oz.     .45  oz.  topwt.    .53  cwt.  to  lb. 


WRITTEN    EXERCISES. 

433.  1.  Reduce  £  bu.  and  .645  da.,  each  to  integers  of 

lower  denominations. 

1st  operation. 

f-bu.X4  =  ^  =  3£pk. 
i  pk.  x  8  =  |  =  2f  qt. 

f  qt.  x2=  -J  =lipt. 

■Jj-bu.  =  3pk.2  qt.l-Jpt. 

1.  The  analyses  of  the  above  are  the  same  as  in  (425)  and  (430). 

2.  The  following  methods  may  be  regarded  as  most  convenient  in 
practice,  since  the  operations  are  performed  without  rewriting  the 
fractional  part  of  each  product. 

2d  operation.  2d  operation. 

5  .645  da. 

4  24 


1st  operation. 
.645  da.  x  24  =  15.48  hr. 
.48  hr.  x  60  =  28.8  min. 
.8  min.  x  60  =  48  sec. 
.645  da.  =  15  hr.  28  min.  48  sec. 


6)20(3  pk.  2  qt.  1\  pt. 
4  18 

1  2 

6  )  8  pt. 8      |  bu. — 3  pk.  2  qt.  1£  P*- 

__6      6 )  1  6  qt. 
2  1JJ 

4 


2580 
1290 

15.4  80hr. 
60 

2  8.8  0  0  min. 
60 


4  8.0  0  0  sec. 
.6  4  5  da.  =  1  5  hr.  2  8  min.  4  8  sec. 


REDUCTION. 


219 


Reduce  to  integers  of  lower  denominations, 


2.  J}  of  a  £. 

3.  .35  lb.  Apoth. 


6.  .625  of  a  fath. 

7.  .55  lb.  Avoir. 


4.  ^  of  a  mi. 

5.  .75  lb.  Troy. 

Eule. — I.  Multiply  the  given  fraction  or  decimal  by 
that  number  in  the  scale  that  will  reduce  it  to  the  next 
lower  denomination.    (425.) 

II.  Proceed  in  like  manner  with  the  fractional  part  of 
each  successive  product,  until  it  is  reduced  to  the  denomi- 
nation required. 

III.  The  integral  parts  of  the  several  products,  arranged 
in  their  proper  order,  is  the  required  result 

Find  the  value  in  integers  of  lower  denominations, 


8.  Off  mo. 

9.  Of  .555£. 

10.  OfTVA. 

11.  Of  |  of  |  lb. 

12.  of  & ca-  yd- 

13.  Of  .1934  S. 

14.  Of  f  =  .7. 


15.  OfJHT. 

16.  Of  .875  hhd. 

17.  Of  $sq.  rd. 

18.  Of  ft>  if 

19.  Of  $  G.  gro. 

20.  Of  .67  lea. 

21.  Of.l25bbl. 


22.  Of  .578125  bu. 

23.  Of  .6625  mi. 

24.  Of  I  of  5£  T. 

25.  Of  {  of  3f  A. 

26.  Of  i  of  3f  Cd. 

27.  Of  {  of  .225  mi. 

28.  Of  .3125  ream. 


29.  At  8-J  cents  a  pound,  what  will  Jf  T.  of  cheese  cost  ? 

434.  To  reduce  denominate  fractions  from  lower 
to  fractions  of  higher  denominations. 

ORAL     EXERCISES. 

1.  Eeduce  f  of  a  peck  to  the  fraction  of  a  bushel. 

Analysis. — Since  4  pk.  make  1  bu.,  there  are  \  as  many  bushels 
as  pecks  ;  i  of  |  pk.  =  {\-  or  *-  bu. 

2.  Eeduce  f  of  a  pint  to  the  fraction  of  a  gallon. 

3.  What  part  of  a  pound  Avoir,  is  f  oz.  ? 

4.  What  part  of  a  week  is  f  da.  ?  -J  da.  ?  f  da.  ?  £  da.  ? 


220  DENOMINATE     HUM  BEES. 

5.  What  decimal  part  of  a  gallon  is  .28  of  a  quart  ? 

Analysis. — Since  4  qt.  make  1  gal.,  there  are  \  as  many  gallons 
as  quarts  ;  \  of  .28  qt.  is  .07  gal. 

6.  Change  .32  of  a  pint  to  the  decimal  of  a  quart. 

7.  What  decimal  of  a  pound  Troy  is  .48  oz.  ?    .84  oz.  ? 

8.  What  decimal  of  a  week  is  .35  da.  ?    .63  da,  ? 

9.  Change  .  72  in.  to  the  decimal  of  a  foot.    Of  a  yd. 

WRITTEN    EXERCISES. 

435.  1.  What  fraction  of  a  bushel  is  f  of  a  pint  ? 

OPERATION.  .  _.    ., 

Analysis.  —  Divide    successive- 

$  pt.  -h  2  =   f   qt.       ly  by  2,  8,  and  4,  the  numbers  in 

^  qt.  -r-  8  =  ^q  pk.      the   ascending   scale,    required   to 

l    w^  _^_  4  _  _jl_  bu.      reduce  pints  to  bushels.    (428.) 

n       m       1       1       1        iv         Hence  -f  pt.  =  -V  DU- 

Or,  fxixixi=-gVbu.  5F       ™ 

2.  Eeduce  ^  of  a  gill  to  the  fraction  of  a  gallon. 

3.  Change  |  of  a  shilling  to  the  fraction  of  a  £. 

4.  Eeduce  .64  of  a  pint  to  the  decimal  of  a  bushel. 

5.  What  part  of  a  pound  Troy  is  .576  of  a  grain  ? 

Eule. — Divide  the  fraction  of  the  lower  denomination 
by  the  numbers  as  factors  in  the  ascending  scale  successively 
between  the  given  and  the  required  denomination.    (428.) 

6.  What  decimal  of  a  ton  is  .8  lb.  ?    .36  of  a  cwt.  ? 

7.  Eeduce  Jf  of  a  cord  foot  to  the  fraction  of  a  cord. 

8.  Eeduce  .216  gr.  to  the  decimal  of  an  ounce  Troy. 

9.  What  part  of  a  ton  is  \  of  a  pound  ? 

10.  What  part  of  a  day  is  £  of  a  minute  ?    .12  hr.  ? 

11.  What  decimal  of  a  rod  are  3.96  in.  ? 

12.  How  much  less  is  |  of  a  pint  than  -^  of  a  hhd.? 

13.  What  part  of  an  acre  is  -£  of  a  square  rod  ? 


REDUCTION.  221 

436.  To  reduce  a  compound  denominate  num- 
ber to  a  fraction  of  a  higher  denomination. 

ORAJL    EXERCISES. 

1.  What  part  of  a  pound  are  4  oz.  ?     8  oz.  ?    10  oz.  ? 

2.  What  part  of  a  foot  are  9  in.  ?    What  part  of  a  yard  ? 

3.  What  part  of  a  bushel  are  2  pk.  4  qt.  ? 

Analysis.— 1  bu.=  32  qt.,  and  2  pk.  4  qt.=  20  qt. ;  20  qt.=f£  bu. 
=  f  bu.,  or  .625  bu.     Hence  2  pk.  4  qt.  =  f  bu.,  or  .625  bu. 

4.  What  part  of  a  yard  are  2  ft.  6  in.  ?    Are  18  in.  ? 

5.  What  part  of  3  lb.  Troy  are  1  lb.  6  oz.  ?    Are  9  oz.  ? 

6.  What  part  of  5  gal.  are  2  gal.  2  qt.  ?     3  gal.  1  qt.  ? 

7.  Eeduce  12  oz.  to  the  decimal  of  3  lb.  Avoir. 

8.  What  fraction  of  3  Cd.  6  cd.  ft.  are  2  Cd.  4  cd.  ft.  ? 

9.  What  part  of  3  pk.  are  1  pk.  4  qt.  ?     Are  2£  pk.  ? 

WRITTEN    EXERCISES. 

437.  1.  What  decimal  of  a  pound  Troy  are  2  oz.  14  pwt.? 
1st  operation.  Analysis. — Since  20  pwt.  make 
20)14  pwt.                              1  oz->  there  are  ^  as  many  ounces 

i  o  \  o  as  Pennv weights ;  and  -^  as  many 

1 2  )  2 .  7  OZ.  pounds  as  ounces  (428).     Hence 

.  2  2  5  lb.  =  ^o  lb.     2  oz.  14  pwt.=.225  lb.,  or  changed 

to  &  fraction,  by  283,  is  ^  lb. 

2d  operation.  Analysis. — In    order   to   find 

2  OZ.  1 4  pwt.  =  54  pwt.  what  part  one  compound  number 

1  ni    9  4.0        r         is  of  another,  both  must  be  like 

"  numbers,  and  must  be  reduced  to 

£4  o  —  4  o  lo.  =  .2  2  O  lb.  the  lowest  denomination  in  either. 

Thus,  2  oz.  14  pwt.  are  equal  to 
54  pwt.,  and  1  lb.  is  equal  to  240  pwt.  Hence  2  oz.  14  pwt.=  /^  lb. 
=  ■&  lb.,  or,  reduced  to  a  decimal,  by  285,  .225  lb. 

2.  Reduce  3  gal.  3  qt.  1-J  pt.  to  the  fraction  of  a  bbl. 

3.  Reduce  3  cd.  ft.  8  cu.  ft.  to  the  decimal  of  a  cord. 


222  DEKOMI^ATE     NUMBERS. 

Rule. — I.  Divide  the  units  of  the  loioest  denomination 
given  by  that  number  in  the  scale  which  is  equal  to  a  unit 
of  the  next  higher  denomination,  and  annex  the  quotient  as 
a  decimal  to  the  number  given  of  that  denomination. 

11.  Proceed  in  like  manner  until  the  whole  is  reduced  to 
the  denomination  required.    Or, 

Reduce  the  given  number  to  its  lowest  denomination  for 
the  numerator  of  the  required  fraction,  and  a  unit  of  the 
required  denomination  to  the  same  denomination  for  the 
denominator,  and  reduce  the  fraction  to  its  lowest  terms, 
or  to  a  decimal. 

1.  If  the  given^  number  contain  a  fraction,  the  denominator  of 
this  fraction  must  be  regarded  as  the  lowest  denomination. 

2.  The  pupil  may  be  required  to  give  the  answers  either  in  the 
form  of  a  fraction,  or  of  a  decimal,  or  both. 

4.  Reduce  13  gal.  3  qt.  3.G2  gi.  to  the  decimal  of  a  hhd. 

5.  What  part  of  a  pound  Troy  are  10  oz.  13  pwt.  9  gr.  ? 
0.  What  fraction  of  2  T.  7  cwt.  28  lb.  are  5  cwt.  91  lb.  ? 

7.  What  part  of  3  A.  80  P.  are  51.52  P.  ? 

8.  What  part  of  a  f  3  are  f  3  5  "l  36  ? 

9.  Change  126  A.  4  sq.  ch.  12  P.  to  the  decimal  of  a  Tp. 
10.  What  decimal  part  of  25°  42'  40"  are  7°  42'  48"  ? 

'  11.  From  a  hhd.  of  molasses  28  gal.  2  qt.  were  drawn. 
What  part  of  the  whole  remained  ? 

12.  What  decimal  of  a  league  are  2  mi.  3  rd.  1  yd.  3f  in.? 

13.  What  part  of  3  bbl.  of  flour  are  110  lb.  4  oz.  ? 

14.  What  decimal  part  of  a  ton  is  \  of  22|  lb.  ? 

15.  Reduce  .45  pk.  to  the  decimal  of  \\  bu. 

16.  What  part  of  54  cords  of  wood  are  4800  cu.  ft.  ? 

17.  Change  18s.  5d.  2^  far.  to  the  fraction  of  a  £. 


REVIEW.  223 

REVIEW. 
WRITTEN     EXAMPLES. 

438.  1.  How  many  steps  of  30  in.  each  must  a  person 
take  in  walking  21  miles  ? 

2.  How  long  will  it  take  one  of  the  heavenly  bodies  to 
move  through  a  sextant,  at  the  rate  of  3'  12"  a  minute  ? 

3.  Reduce  £10  18s.  6d.  to  United  States  Money. 

4.  Paid  $425.75  for  2^  tons  of  cheese,  and  retailed  it  at 
12£  cents  a  pound.     What  was  the  whole  gain  ? 

5.  Reduce  580  francs  to  United  States  Money. 

6.  Change  $291.99  to  Sterling  Money. 

7.  What  cost  30  bu.  2  pk.  1  qt.  of  beans,  at  $4.20  a  bu.  ? 

8.  Bought  15  cwt.  22  lb.  of  rice  at  $4.25  a  cwt.,  and  6 
cwt.  36  lb.  of  pearl-barley  at  $5.60  a  cwt.  What  would  be 
gained  by  selling  the  whole  at  6£  cents  a  pound  ? 

9.  How  many  bushels  of  corn  in  36824  lb.,  Illinois 
standard?    Louisiana?    New  York? 

10.  5000  bu.  of  oats  in  Ohio  are  equal  to  how  many 
bushels  in  Connecticut,  by  weight  ?    In  New  Jersey  ? 

11.  If  I  buy  16  T.  3  cwt.  3  qr.  24  lb.  of  Eng.  iron,  by 
long  ton  weight,  at  3d.  a  lb.,  and  sell  the  same  at  $140, 
by  the  short  ton,  what  do  I  gain  by  the  transaction  ? 

12.  How  many  carats  fine  is  a  piece  of  gold  -f  pure  ? 

13.  How  many  acres  in  a  piece  of  land  105  ch.  85  1. 
long,  and  40  ch.  15  1.  wide  ? 

14.  If  10  lb.  of  milk  make  1  lb.  of  cheese,  what  will  it 
cost  at  1  cent  a  pound  to  manufacture  the  cheese  that 
can  be  made  from  90000  lb.  of  milk  ? 

15.  At  $75f  an  acre,  what  is  the  value  of  a  farm  189.5 
rd.  long  and  150  rd.  wide? 


224  DENOMINATE     NUMBERS. 

16.  What  cost  2  bu.  3  pk.  6  qt.  of  green  peas,  at  $.30  a 
peck? 

17.  What  cost  3  T.  17  cwt.  20  lb.  of  hay,  at  $22|  a  ton  ? 

18.  If  a  grocer's  scales  give  \  oz.  short  of  true  weight 
on  every  pound,  of  how  much  money  does  he  defraud  his 
customers,  in  the  sale  of  3  bbl.  of  sugar,  each  weighing  2 
cwt.  10  lb.,  at  12^  cents  a  pound  ? 

19.  If  37  A.  128  P.  are  sold  from  a  farm  containing 
170  A.  16  P.,  what  part  of  the  whole  remains  ? 

20.  Paid  $526.05  for  3-J-  tons  of  cheese,  and  retailed  it 
at  9  J  cents  a  pound.     How  much  was  the  whole  gain  ? 

21.  How  many  bushels  of  oats  in  Connecticut  are 
equivalent  in  weight  to  2500  bushels  in  Iowa  ? 

22.  How  many  centals  of  barley 'in  California  are  equiv.. 
alent  to  1500  bushels  in  Missouri  ? 

23.  A  man  sold  12  bu.  3  pk.  6  qt.  of  cranberries  at  $3£ 
a  bushel,  and  took  his  pay  in  flour  at  4  cents  a  pound. 
How  many  barrels  did  he  receive  ? 

24.  If  3  T.  12  cwt.  20  lb.  of  ground  plaster  cost  $15.75, 
what  will  be  the  cost  of  5  T.  80  lb.  at  the  same  rate  ? 

25.  Bought  37  Cd.  48  cu.  ft.  of  wood  for  $129.81,  and 
there  was  but  13  Cd.  59  cu.  ft.  delivered.  What  part  of 
the  money  should  be  paid  ? 

26.  If  a  grocer's  gallon  measure  is  too  small  by  1  gi., 
what  does  he  make  dishonestly  in  selling  2  hhd.  of  mo- 
lasses, averaging  58  gal.  2  qt.  1  pt.  each,  worth  $.80  a  gal.? 

27.  How  many  reams  of  paper  are  required  to  supply 
4500  subscribers  with  a  weekly  newspaper  for  1  year  ? 

28.  A  publisher  printed  an  edition  of  10000  copies  of  a 
12mo  book  of  336  pp.  How  much  paper  did  he  use, 
allowing  1  quire  to  each  ream  for  waste  ? 


ADDITION.  225 

ADDITION. 

439.  Denominate  numbers  are  added,  subtracted,  mul- 
tiplied, and  divided  by  the  same  general  methods  as  are 
employed  for  like  operations  in  Simple  Numbers. 

The  corresponding  processes  are  based  upon  the  same 

principles.     The  only  modification  of  the  rules  needed  is 

that  which  is  required  by  a  varying  scale  instead  of  a 

uniform  scale  of  10. 

The  principles  will  be  made  sufficiently  plain  in  the  operations 
and  analyses  to  render  special  rules  unnecessary. 

WRITTEN     EXERCISES. 

440.  1.  Find  the  sum  of  4  cwt.  46  lb.  12  oz.,  12  cwt. 

9|  lb.,  2i  cwt.,  and  21f  lb. 

Analysis. — Write  the  numbers  so  that  units 
of  the  same  denomination  stand  in  the  same 
column,  and  begin  at  the  right  to  add. 

The  sum  of  the  ounces  is  30,  equal  to  1  lb. 
14  oz.  Write  the  14  oz.  under  the  column  of 
ounces,  and  add  the  1  lb.  to  the  pounds  of  the 
next  column. 

19  2  The  sum  of  the  pounds  is  102  lb.,  equal  to  1 

cwt.  and  2  lb.     Write  the  2  lb.  under  the 

column  of  pounds,  and  add  the  1  cwt  to  the  cwt  of  the  next  column. 

The  sum  of  the  cwt.  is  19  cwt.,  which  write  under  the  column  of 

cwt.     Hence  the  entire  sum  is  19  cwt.  2  lb.  14  oz. 

2.  What  is  the  sum  of  ^  wk.,  |  da.,  and  f  hr.  ? 

1st  operation.  Analysis. — First  find  the 

da.       hr.       min.      sec.      value    of    each    denominate 

7    wk#  —  4       21        36        00      fraction  in  integers  of  lower 

o-i„  -,  A        «  a        (\f\      denominations    (433),    and 

4  da.    =  14      2  4      0  0  ,.        v 

0  then  add  the  resulting  com- 

i   nr#    —  ^*       3Q     pound  numbers.     Or, 

5      12      22      30 


OPERATION. 

cwt. 

lb. 

oz. 

4 

46 

12 

12 

9 

8 

2 

25 

0 

21 

10 

226  DENOMINATE     NUMBERS. 

2d  operation.  Reduce    the    given 

s  A*  __    3   wv  fractions  to  fractions  of 

the  same  denomination 
|  hr.  =  fa  wk.  (434),  then  add  the  re- 

TV  wk.  +  -3%  +  xh"  =  Iff  Wk.  suits  and  find  the  value 

fjf  wk.  =  5  da.  12  hr.  22  min.  30  sec.     of  their  sum  in  integers 

of  lower  denominations. 

If  denominate  fractions  occur  in  the  given  numbers,  they  should 

be  reduced  to  integers  of  lower  denominations  (433)  before  adding. 

3.  Add  7  yd.  2  ft.,   5  yd.  1±  ft.,  2  ft.  9$  in.,   3  yd.  1  ft. 
6$  in.,  2}  ft.,   and4£yd. 

4.  Add  5  Cd.  7  cd.  ft.,  2  Od.  2  cd.  ft.  12  cu.  ft.,   6  cd.  ft. 
15  cu.  ft.,   7|  Od.,   and  3  Cd.  2  cu.  ft. 

5.  What  is  the  sum  of  If  hhd.,    36  gal.  3  qt.  1J  pt., 
J  gal.,    2  qt.  }  pt.,  and  1.75  pt.  ?    • 

6.  What  is  the  sum  of  f  of  a  day  added  to  ^  of  an  hour  ? 

7.  To  f  of  a  hhd.  add  £  of  10  gal. 

8.  What  is  the  sum  of  22^-  cwt,   26-J  lb.,  and  14  oz.  ? 

9.  Add  5£  Pch.,    18  cu.  ft.,    86.6  cu.  ft.,  and  f-  Pch. 

10.  Find  the  sum  of  lb  4  §  6  3  5,  and  ft)  6  3  4£  3  9J. 

11.  A  Missouri  farmer  received  $.75  a  bushel  for  4  loads 
of  corn  ;  the  first  contained  48.4  bu.,  the  second  2626  lb., 
the  third  36f  bu.,  and  the  fourth  41  bu.  52  lb.  What 
did  he  receive  for  the  whole  ? 

12.  Bought  three  loads  of  hay  at  $15  a  ton.  The  first 
weighed  1.125  T.,  the  second  If  T.,  and  the  third  2750  lb. 
What  did  the  whole  cost  ? 

13.  When  B  was  born,  A's  age  was  3  yr.  9  mo.  24  da. ; 
when  0  was  born,  B's  age  was  12  yr.  19  da. ;  when  D  was 
born,  C's  age  was  5  yr.  11  mo.,  and  when  E  was  born, 
D's  age  was  10  yr.  1  mo.  20  da.  What  was  A's  age  when 
E  was  born  ? 


4  yd. 

11.6  in. 

OPERATION. 

rd. 

yd.         ft. 

in. 

25 

2        2 

6.3 

12 

4        0 

11.6 

12 

3(i)l 

6.7 

f  =  1 

6 

12        4        0  .7 


SUBTRACTION.  227 

SUBTRACTION. 

WRITTEN    EXERCISES. 

441.  1.  From  25  rd.  2  yd.  2  ft.  6.3  in.,  subtract  12  rd. 

Analysis. — Write  the  numbers, 
so  that  units  of  the  same  denomina- 
tion stand  in  the  same  column,  and 
begin  at  the  right  to  subtract. 

Since  11 .6  in.  cannot  be  subtracted 
from  6.3,  take  1  ft,  equal  to  12  in., 
from  the  2  ft.,  leaving  1  ft.,  and  add 
it  to  the  C.3  in.,  making  18.3  in. 
Subtract  11.6  in.,  and- write  the  re- 
mainder, 6.7  in.,  under  the  inches. 
Since  1  ft.  has  been  taken  from  2  ft.,  subtract  0  ft.  from  1  ft., 
and  write  the  remainder  1  ft.  under  the  feet. 

Since  4  yd.  cannot  be  taken  from  2  yd.,  take  1  rd.,  equal  to  5|  yd., 
from  25  rd.,  leaving  21  rd.,  and  add  it  to  the  2  yd.,  making  7£  yd. 
Subtract  4  yd.  from  lh  yd.,  and  write  the  remainder,  3|  yd.,  under 
the  yards. 

Since  1  rod  has  been  taken  from  25  rd.,  subtract  12  rd.  from  24  rd., 
and  write  the  remainder,  13  rd.,  under  the  rods. 

The  |  yd.,  reduced  to  feet  and  inches,  and  added  to  1  ft.  6.7  in. 
of  the  remainder,  gives  12  rd.  4  yd.  .7  in. 

2.  From  If  bu.  subtract  %  bu. 

operation.  Analysis.  —  First 

If  bu.  =  lbu.  2  pk.  4  qt.  0  pt.       find  tne  value  of  eacQ 

denominate  fraction  in 
integers  of  lower  de- 
3  2^  nominations     (433), 

and  subtract  the  less 
value  from  the  greater. 
If  bu.=^bu.;  J^bu.—  f  bu.=£J  bu.        Or,  reduce  the  given 
J^-  bu.  —  3  pk.  2  qt.  ^  pt.  fractions  to   fractions 

of  the  same  denomi- 
nation (434),  then  subtract  the  less  from  the  greater,  and  find  the 
value  of  their  difference  in  integers  of  lower  denominations. 


ibu.  = 3_ 

3 
Or, 


228  DENOMINATE     NUMBEES. 

3.  From  a  pile  of  wood  containing  42  Cd.  5  cd.  ft., 
take  16  Cd.  6  cd.  ft.  12  cu.  ft.,  and  how  much  remains  ? 

4.  From  the  sum  of  f  of  3 J  mi.  and  17|  rd.,  take 
120£  rd. 

Find  the  difference  between 


10.  -f  lea.  and  -fo  mi. 

11.  f  gross  and  f  doz. 

12.  3^31  and  §  4. 

13.  .9  da.  and  \  wk. 

14.  ^  A.  and  84.56  P. 


5.  8^  cwt.  and  48$  lb. 

6.  £f  andf  offs. 

7.  Alb-and5lb-4oz-8Pwfc- 

8.  .659  wk.  and  2  wk.  3|  da. 

9.  m  hhd.  and  3.625  gal. 

15.  If  from  a  hhd.  of  molasses  14  gal.  1  qt.  1  pt.  be 
drawn  at  one  time,  10  gal.  3  qt.  at  another,  and  29  gal. 
1  pt.  at  another,  how  much  will  remain  ? 

16.  Of  a  farm  containing  250  A.,  two  lots  were  re- 
served, one  containing  75  A.  136.4  P.,  and  the  other 
56  A.  123.3  P. ;  the  remainder  was  sold  at  $62J-  an  acre. 
What  did  it  sell  for? 

17.  From  1  T.  11  cwt.  30  lb.,  take  £  of  a  long  ton. 

18.  From  a  pile  of  wood  containing  125f  Cd.,  was  sold 
at  one  time  26  Cd.  7  cd.  ft.  ;  at  another,  30  Cd.  4|  cd.  ft; 
at  another,  37^  Cd.    How  much  remained  ? 

442.  To  find  the  interval  of  time  between  two 
dates. 

1.  How  many  yr.,  mo.,  da.,  and  hr.,  from  3  o'clock  p.  M. 
of  May  16,  1864,  to  9  o'clock  a.m.  of  Sept.  25,  1875  ? 

operation.  Analysis. — Since  the   later   date 

yr.  mo.  da.  hr.  expresses  the  greater  period  of  time, 
1875        9        25  9        write  li  as  tne  minuend,  and  the  ear- 

i  r      i  ^      lier  date  ^  tlie  sul3tranend'  writing 

1864       5        lo        10       the  denominations  in  the  order  of  the 
11       4  8       18       scale,  then  subtract. 


SUBTRACTION.  229 

1.  When  hours  are  to  be  obtained,  reckon  from  10  at  night,  and 
if  minutes  and  seconds,  write  them  still  at  the  right  of  hours. 

2.  In  finding  the  difference  of  time  between  two  dates,  12  mo.  are 
usually  considered  a  year,  and  30  days  a  month. 

3.  When  the  time  is  less  than  a  year,  the  true  number  of  days  in 
each  month  and  parts  of  a  month  is  added. 

4.  The  day  on  which  a  note,  draft,  or  contract  is  dated,  and  that 
on  which  they  mature,  are  not  both  included.  The  former  is  gen- 
erally omitted. 

2.  The  war  between  England  and  America  was  com- 
menced April  19,  1775,  and  peace  was  restored  Jan.  20, 
1783.     What  length  of  time  did  the  war  continue  ? 

3.  The  American  Civil  War  began  April  11,  1861,  and 
closed  April  9,  1865.     What  time  did  it  continue  ? 

4.  How  long  has  a  note  to  run  that  is  dated  Jan.  16, 
1873,  and  made  payable  July  10,  1875  ? 

5.  A  note  dated  May  28,  1875,  was  paid  Feb.  10, 1876. 
What  length  of  time  did  it  run  ? 

6.  A  person  started  on  a  tour  of  the  world  at  9  o'clock 
A.  m.,  Sept.  3,  1874,  and  returned  to  the  same  depot  at 
3  P.  m.,  July  15,  1876.     What  time  was  he  absent? 

7.  How  many  years,  months,  and  days  from  your  birth- 
day to  this  date;  or,  what  is  your  age? 

8.  How  many  days  from  June  20th  to  the  10th  of  Jan. 
following  ? 

9.  What  length  of  time  elapsed  from  12  o'clock  m., 
Jan.  10,  1876,  to  June  16,  9  o'clock  a.m.  ? 

10.  What  length  of  time  elapsed  from  16  min.  past  10 
o'clock  a.m.,  July  4,  1873,  to  22  min.  before  8  o'clock 
p.m.,  Dec.  12,  1875? 

11.  What  length  of  time  will  elapse  from  40  min. 
25  sec.  past  12  o'clock  m.,  April  21,  1875,  to  4  min. 
36  sec.  before  5  o'clock  p.m.,  Jan.  1,  1878  ? 


230  DEKOMINATE     NUMBEBS. 

MULTIPLICATION". 

WRITTEN     EXERCISES. 

443.  1.  Multiply  28  rd.  2  yd.  2  ft.  by  7. 

operation.  Analysis. — Write   the   multi- 

2  8  rd.      2  yd.      2  ft.  P^ier  un(ler  the  lowest  denominar 

m  tion    of    the    multiplicand,    and 

multiply. 


199  1  (;L)      2  7  times  2  ft.  are  14  ft.,  equal  to 

|-=1        6  in.      4  yd.  2  ft.     Write  the  2  ft.  under 

1QQ  ~  7T        T"  the  feet,  and  reserve  the  4  yd.  to 

add  to  the  product  of  yards. 

7  times  2  yd.  are  14  yd.,  and  4  yd.  added  make  18  yd.,  equal  to 
3  rd.  1^  yd.  Write  the  1\  yd.  under  the  yards,  and  reserve  the  3 
rd.  to  add  to  the  product  of  rods. 

7  times  28  rd.  are  196  rd.,  and  3  rd.  added  make  199  rd.,  which 
write  under  the  same  denomination. 

The  I  yd.  is  equal  to  1  ft.  6  in.,  which  added  to  the  product,  gives 
199  rd.  2  yd.  6  in.  for  the  entire  product. 

1.  The  multiplier  must  be  an  abstract  number.     (103.) 

2.  When  the  multiplier  is  large  and  is  a  composite  number,  the 
work  may  be  shortened  by  multiplying  successively  by  its  factors. 
(109.) 

2.  In  9  bbl.  of  walnuts,  each  containing  2  bu.  3  pk. 
6  qt.,  bow  many  bushels? 

3.  If  a  man  cut  3  Cd.  36  cu.  ft.  of  wood  in  1  da.,  how 
many  cords  can  he  cut  in  12  days  ? 

4.  Multiply  8  gal.  3  qt.  1  pt.  3.25  gi.  by  96. 

5.  If  1  A.  produce  42  bu.  1  pk.  5  qt.  1  pt.  of  corn,  how 
many  bushels  will  64  A.  produce? 

6.  Multiply  0.  8  f  3  9  f  3  6  ni  34  by  24. 

7.  What  will  84  yd.  of  cloth  cost,  at  £1  8s.  9-Jd.  a  yd.? 

8.  If  $80  will  buy  3  A.  24  P.  20  sq.  yd.'  4  sq.  ft.  of  land, 
how  much  will  $4800  buy  ? 


24  bu. 

Opk. 

6  qt.  in  9  bags. 
5 

±zQ  bu. 
•5 

3pk. 
1 

6  qt.  "45     " 
4        "    2     " 

MULTIPLICATION.  231 

9.  How  many  bushels  of  grain  in  47  bags,  each  con- 
taining 2  bu.  2  pk.  6  qt.  ? 

opekation.  Analysis.— Multiplying 

4.7  _  /g  x  §\    I    2  tne  contents  of  1  bag  by  9, 

and  the  resulting  product 
Z  bu.     2  pk.     6  qt.  Dy  5}  gives  t]ie  contents  of 

9  45  bags,  which  is  the  com- 

posite number  next  less 
than  the  given  prime  num- 
ber, 47.  Next  find  the 
contents  of  2  bags,  which, 
added  to  the  contents  of  45 
bags,  gives  the  contents  of 
126  bu.     lpk.     2qt.  -47     "         45  +  2  or  47  bags. 

10.  If  a  load  of  coal  by  the  long  ton  weigh  1  T.  6  cwt. 
2  qr.  26  lb.  10  oz.,  what  will  be  the  weight  of  67  loads? 

11.  Multiply  4  yd.  1  ft.  4.7  in.  by  125. 

12.  Multiply  7  T.  15  cwt.  10.5  lb.  by  1.7. 

13.  At  $1.37£  a  gallon,  what  will  be  the  cost  of  5  casks 
of  wine,  each  containing  28  gal.  2  qt.  1  pt.  ? 

14.  A  farmer  sold  4  loads  of  oats,  averaging  41  bu.  3  pk. 
each,  at  $.  75  a  bushel.   What  did  he  receive  for  the  whole  ? 


DIVISION. 

444.  1.  Divide  56  lb.  9  oz.  12  pwt.  by  6. 

opekation.  Analysis. — Write  the  divisor  at  the  left 

lb.        oz.     pwt.      of  the  dividend.     The  object  is  to  find   1 
6)56        9        12       *&**  of  a  compound  number. 

o       k       To  %  of  ^  lb*  is  ®  ^'  an(*  a  remam<ler  of  2  lb. 

Write  the  9  ib  m  thg  quotient,  and  reduce 
the  2  lb.  to  ounces,  which,  added  to  9  oz.,  make  33  oz. 

-J-  of  33  oz.  is  5  oz.  and  a  remainder  of  3  oz.  Write  the  5  oz.  in 
the  quotient,  and  reduce  the  3  oz.  to  pwt.,  which  added  to  12  pwt., 
make  72  pwt.     £  of  72  pwt.  is  12  pwt.,  which  write  in  the  quotient. 


232  DENOMINATE     NUMBERS. 

2.  Divide  358  A.  57  P.  6  sq.  yd.  2  sq.  ft.  by  7. 

3.  Divide  £35  9s.  7d.  by  5  ;  by  7  ;  by  8. 

4.  Divide  282  bu.  3  pk.  1  qt.  1  pt.  by  9  ;  by  10  ;  by  12. 

When  the  divisor  is  large,  and  is  a  composite  number,  the  work 
may  be  shortened  by  dividing  successively  by  its  factors. 

5.  Divide  254  yd.  4  ft.  3£  in.  by  21  ;    by  42. 

6.  Divide  196  Cd.  4  cd.  ft.  12  cu.  ft.  by  72. 

7.  How  many  iron  rails,  each  16  ft.  long,  are  required 
to  lay  a  railroad  track  26  mi.  long  ? 

8.  Divide  24  sq.  mi.  140  P.,  by  22J. 

9.  Divide  202  yd.  1  ft.  6f  in.  by  |. 

10.  Divide  336  bu.  3  pk.  4  qt.  by  4  bu.  3  pk.  2  qt. 

Reduce  both  dividend  and  divisor  to  the  same  denomination,  and 
divide  as  in  simple  numbers. 

11.  How  many  boxes,  each  holding  1  bu.  1  pk.  7  qt., 
can  be  filled  from  356  bu.  3  pk.  5  qt.  of  cranberries  ? 

12.  Divide  311  gal.  1  qt.  1  pt.  by  53. 

OPERATION. 

53)311  gal.  1  qt.  1  pt.(5  gal.  3  qt.  1  pt. 
265 

4  6  gal.  rem.  2  6  qt.  rem. 

4  2 


185  qt.  in  46  gal.  1  qt.  53  pt.  in  26  qt.  1  pt. 

159  63 

2  6  qt.  rem. 

13.  The  aggregate  weight  of  41  hhd.  of  sugar  is  19  T. 
6  cwt.  22  lb.     What  is  the  average  weight  ? 

14.  If  a  town  4  mi.  square  be  equally  divided  into  62 
farms,  how  much  land  will  each  farm  contain  ? 


LONGITUDE     AND     TIME.  233 

LONGITUDE    AND    TIME. 

445.  The  Longitude  of  a  place  is  its  distance  east 
or  west  from  a  given  meridian,  measured  on  the  equator. 

The  meridian  from  which  longitude  is  reckoned  is 
called  the  first  meridian,  and  is  marked  0°.  All  places 
east  of  this,  within  180°,  are  in  east  longitude,  and  all 
places  west,  within  180°,  are  in  west  longitude. 

The  English  and  Americans  usually  reckon  longitude  from  the 
meridian  of  Greenwich,  England  ;  the  French,  from  Paris. 

446.  Since  the  earth  revolves  on  its  axis  once  in  24 
hours,  the  sun  appears  to  pass  from  east  to  west  around 
the  earth,  or  through  360°  of  longitude  once  in  24  hours 
of  time.  Hence  in  1  hour  the  sun  appears  to  pass  through 
^  of  360°,  or  15°  ;  in  1  minute,  through  -fa  of  15°,  or 
15' ;  and  in  1  second,  through  -fa  of  15',  or  15". 

Comparison  of  Longitude  and  Time. 


''or  a  difference  of 

There  is  a  difference  of 

15°  in 

Long. 

1  hr.    in 

Time. 

15'    " 

tt 

1  min.  " 

" 

15"  " 

tf 

1  sec.    " 

tt 

1°    " 

<( 

4  min.  " 

n 

1'    " 

" 

4  sec.    " 

a 

1"  " 

tt 

TVsec.   " 

<t 

1.  Since  the  sun  appears  to  move  from  east  to  west,  when  it  is 
12  o'clock  at  one  place,  it  will  be  past  12  o'clock  at  all  places  east, 
and  before  12  at  all  places  west.  Hence,  knowing  the  difference 
of  time  between  two  places,  and  the  exact  time  at  one  of  them, 
the  exact  time  at  the  other  is  found  by  adding  their  difference  to 
the  given  time,  if  it  is  east,  and  by  subtracting,  if  it  is  west 

2.  If  one  place  is  in  east  and  the  other  in  west  longitude,  the 
difference  of  longitude  is  found  by  adding  them,  and  if  the  sum  is 
greater  than  180°,  by  subtracting  it  from  360°. 


234  DENOMINATE     NUMBERS. 

ORAL    EXERCISES. 

447.  1.  The  earth  revolves  on  its  axis  once  in  every 
24  hr.    What  part  of  a  revolution  does  it  make  in  12  hr.? 

2.  How  many  degrees  of  the  earth's  surface  pass  under 
the  sun's  rays  in  24  hr.  ?   In  12  hr.  ?   In  4  hr.  ?  In  1  hr.  ? 

3.  How  many  degrees  of  longitude  cause  a  difference 
of  1  hr.  in  time  ?     2  hr.  ?     3  hr.  ? 

4.  When  it  is  6  o'clock  at  Chicago,  what  is  the  hour 
15°  east  of  Chicago  ?    15°  west  of  Chicago  ? 

5.  When  it  is  noon  in  New  York,  what  is  the  hour  15° 
east  of  New  York  ?    30°  west  of  1ST.  Y.  ? 

6.  When  it  is  3  o'clock  at  Washington,  what  is  the 
time  15°  15'  east  of  Washington  ?    30°  30'  west  ? 

7.  What  difference  of  longitude  causes  a  difference  of 
1  hr.  in  time  ?     Of  1  minute  ?     Of  1  second  ? 

8.  If  the  difference  in  the  time  of  Boston  and  of  St.  Louis 
is  1  hr.  15  min.,  what  is  the  difference  in  their  longitude  ? 

9.  A  man  left  New  Orleans  and  traveled  until  his 
watch  was  1  hr.  2  min.  too  fast.  How  far  had  he  trav- 
eled, and  in  what  direction  ? 

10.  Two  persons,  at  different  points,  observe  an  eclipse 
of  the  moon  ;  one  seeing  it  at  9£  P.  M.,  and  the  other  at 
midnight.     What  is  the  difference  in  their  longitude  ? 

11.  A  tourist  leaves  home  at  12  m.  on  Monday,  and  on 
Saturday  finds  his  watch  1  hr.  15  min.  slow.  In  what 
direction  has  he  been  traveling  ?    How  far  ? 

12.  A  and  B  start  from  opposite  points  and  travel 
towards  each  other.  When  they  meet,  A's  watch  is  40 
min.  slow  and  B's  1  hr.  fast.  How  far  apart  are  the  two 
points  of  starting,  and  in  what  direction  did  each  travel  ? 


52 

56 
15 

Diff.  in 
Diff.  in 

Time. 

13° 

14' 

0  0" 

Long. 

Or,     4) 

5  2  min.     5  G 

sec. 

LONGITUDE     AND     TIME.  235 

WRITTEN      EXERCISES. 

448.  To  find  the  difference  of  longitude  between 
two  places,  when  the  difference  of  time  is  known. 

1.  When  it  is  9  o'clock  at  Washington,  it  is  7  min.  4  sec. 
past  8  o'clock  at  St.  Louis.     Find  the  diff.  of  longitude. 

Analysis. — Since  every 

operation.  hour  of  time  corresponds 

9  hr.   0  min.  0  sec.  to  15°  of  long.,  and  every 

8  7  4  minute  of  time  to  15'  of 

long.,  and  every  second  of 

time  to  15"  of  long.  (446), 

there  are  15  times  as  many 

deg.,  min.,  and  sec.  in  the 

difference  of  longitude,  as 

there  are  hr.,   min.,    and 

13°  14'  sec.  in  the    difference  of 

time.     Or, 

Since  4  min.  of  time  make  a  difference  of  1°  of  long.,  and  4  sec. 

of  time  a  difference  of  1'  of  long.,  there  will  be  \  as  many  degrees 

of  long,  as  there  are  minutes  of  time,  and  J  as  many  minutes  of 

long,  as  there  are  seconds  of  time. 

2.  The  difference  in  the  "time  of  Washington  and  of  St. 
Petersburgh  is  7  hr.  9  min.  19}  sec.  What  is  the  differ- 
ence in  their  longitudes  ? 

3.  When  it  is  12  o'clock  M.  at  Eochester,  N.  Y.,  it  is 
9  hr.  1  min.  37  sec.  A.  M.  at  San  Francisco.  The  long,  of 
Eochester  being  77°  51'  W. ,  what  is  the  long,  of  the  latter  ? 

Eule. — Multiply  the  difference  of  time  expressed  in 
hours,  minutes,  and  seconds  by  15  ;  the  product  will  he  the 
difference  of  longitude  in  degrees,  minutes,  and  seconds.    Or, 

Reduce  the  difference  of  time  to  minutes  and  seconds, 
then  divide  by  4  ;  the  quotient  will  be  the  difference  of  lon- 
gitude in  degrees  and  minutes. 


236 


DENOMINATE     NUMBERS 


4.  Noon  comes  1  lir.  5  min.  42  sec.  sooner  at  Quebec 
than  at  Chicago,  whose  longitude  is  87°  37'  45".  What 
is  the  longitude  of  Quebec  ? 

5.  When  the  days  and  nights  are  of  equal  length,  and 
it  is  noon  on  the  first  meridian,  on  what  meridian  is  it 
then  sunrise  ?     Sunset  ?     Midnight  ? 


449.  Tlie  folloiving  table  of  the  Longitude  of  places  is 
compiled  from  the  Records  of  the  U.  &.  Coast  Survey. 


Albany 

Ann  Arbor 

Astoria,  Or 

Boston 

Berlin 

Bombay 

Cincinnati 

Chicago. 

Cambridge,  Mass. 
Jefferson  City,  Mo. 
Mexico 


73°  44' 
83°  43' 
124° 
71°  3' 
13°  23' 
72D  54' 
84°  29' 
87°  37' 
71°  7' 
92°  8' 
99°    5' 


50"  W. 
W. 

w. 

30  "W. 
45"  E. 
E. 
31"  W. 

45"  W. 

40"  W. 

W. 

w. 


New  York 

New  Orleans 

Paris 

Rome 

Richmond,  Va... 

San  Francisco 

St.  Paul,  Minn... 
St.  Louis,  Mo. . . 
Univ.  of  Virginia. 
West  Point,  N.  Y. 
Washington,  D.  C. 


74°  3' 
90°  2' 
2°  20' 
12°  27' 
77°  25' 
122°  26' 
95°  4' 
90°  15' 
78°  31' 
73°  57' 
77°   0' 


W. 

30"  W. 

E. 

E. 
45"  W. 
45"  W. 
55  W. 
15"  W. 
30  "W. 

W. 
15"  W. 


450,  To   find   the   difference   of   time    between 
two  places,  when  their  longitudes  are  given. 

1.  Find  the  diff.  in  the  time  of  Cinn.  and  of  St.  Panl. 


9  5< 
84 


4' 
29 


OPERATION. 

5  5"    Long. of  St.  P. 
31         "  Cinn. 


Analysis.  — 
Since  15°of  lon- 
gitude make  a 
difference  of  1 
hr.  of  time,  and 
15',  a  difference 
of  1  min.  of  time, 

and  15",  a  difference  of  1  sec.  of  time  (44G),  there  are  fs  as  many 
hours,  minutes,  and  seconds  of  time  as  there  are  degrees,  minutes, 
and  seconds  of  longitude. 


1  5  )  10°    35'         2 4"    Diff.  of  Long. 

4  2  min.  2 1  f  sec.  Diff.  of  Time. 


,ONGITUDE 

Atf  D 

TIME. 

10° 

3  5' 

2  4" 

4 

rain. 
=  42 

sec. 

4  2  rain. 

2 1  sec.     -Jf 

81| 

23? 


Or, 


Since  1°  of  long,  makes  a  diff.  of  4  min.  of  time,  and  V  makes  a 
diff.  of  4  sec.  of  time  (446),  there  is  a  diff.  of  4  times  as  many  mim 
utes  and  seconds  of  time  as  there  are  deg.,  min.,  and  sec.  of  long. 

2.  Find  the  difference  in  the  time  of  Ann  Arbor, 
Mich.,  and  of  Cambridge,  Mass  ?  * 

3.  When  it  is  half-past  3  o'clock  p.m.  at  West  Point, 
N.  Y.,  what  time  is  it  at  Bombay  ? 

Rule. — Divide  the  difference  of  longitude  expressed  in 
degrees,  minutes,  and  seconds,  by  15  ;  the  quotient  ivill  he 
the  difference  of  time  in  hours,  minutes,  and  seconds.     Or, 

Multiply  the  difference  of  longitude  by  4,  and  the  product 
will  be  the  difference  of  time  in  minutes  and  seconds,  which 
may  be  reduced  to  hours. 

Find  the  difference  in  the  time  of 


4.  Washington,  and  Rome. 

5.  Chicago,  and  Paris. 

6.  N.  Orleans,  and  N.  York. 

7.  Albany,  and  Jefferson  C'y. 


8.  Richm'd,  and  St.  Louis. 

9.  New  York,  and  Mexico. 

10.  Ann  Arbor,  and  Berlin. 

11.  Mexico,  and  San  Fran. 


12.  When  it  is  6  a.m.  at  Boston,  what  time  is  it  at  Cin- 
cinnati ?    At  Chicago  ?    At  St.  Louis  ? 

13.  When  it  is  6  p.m.  at  the  University  of  Va.,  what 
time  is  it  at  Berlin  ?    At  St.  Paul  ?    At  Astoria,  Or.  ? 

14.  How  much  later  does  the  sun  rise  in  New  York 
than  in  Rome  ?    Than  in  Paris  ? 

15.  In  sailing  from  San  Francisco  to  Bombay,  will  a 
chronometer  gain  or  lose  time,  and  how  much  ? 

*  Take  from  the  Table  the  required  Longitude  of  the  different  places. 


238  DENOMINATE     NDMBEBS. 

DUODECIMALS. 

451.  Duodecimals  are  fractions  of  a  foot  formed 
by  successively  dividing  by  1 2  ;  as,  -^  T£4-,  -pjVs,  e^c- 

452.  The  Unit  of  measure  is  1  foot,  which  may  be  a 
linear,  a  square,  or  a  cubic  foot.    The  scale  is  uniformly  12. 

453.  In  the  duodecimal  divisions  of  a  foot,  the  differ- 
ent orders  of  units  are  related  as  follows  : 

1'    (inch  or  prime)  =    ifa    of  a  foot,  or  1  in.  Linear  Meas. 

1"  (second)  or  fj  of  -fa  =  ji*  of  a  foot,  or  1  "  Square     " 

V"  (third)  or  ^  of  -fa  of  -h  =  Wrs  of  a  foot,  or  1  "  Cubic 


Table. 


12  Fourths  ("")=1  Third    .  .  1" 

12  Thirds  =1  Second  .  .  1" 

12  Seconds        =1  Prime    .  .   V 

12  Primes         =1  Foot  .   .  .  ft. 


1/*.=12'=144"=1728'"=20736"" 

1'=  12"=  144'"=  1728"" 

1"=     12'"=     144"" 

1'"=      12"" 


The  marks  ',  ",  '",  "",  are  called  Indices. 

Duodecimals  are  used  by  artificers  in  measuring  surfaces  an<?  solids. 


ADDITION    AND    SUBTRACTION. 

454.  Duodecimals  are  added  and  subtracted  i**  the 
same  manner  as  compound  numbers. 

WRITTEN     EXERCISES. 

1.  Add  14  ft.  7'  8",  16  ft.  3'  5",  and  21  ft.  9'  11". 

2.  Add  140  ft.  10'  7"  9'",  71  ft.  8",  and  107  ft.  4'  11"  3'". 

3.  From  54  ft.  9'  5"  subtract  30  ft.  10'  8". 

Duodecimals  are  not  much  used.    The  subject  is  fully  treated 
and  applied  in  "  Robinson's  Higher  Arithmetic." 


9  ft. 

4  ft. 

8' 

r 

5  ft. 

3  8  ft. 

7    8" 
8' 

DUODECIMALS.  239 

MULTIPLICATION. 

455.  In  the  multiplication  of  duodecimals,  the  product 
of  two  dimensions  is  area  or  surface,  and  the  product  of 
three  dimensions  is  solidity  or  volume.     (344,  349.) 

WRITTEN     EXERCISES. 

456.  1.  Multiply  9  ft.  8'  by  4  ft.  7. 

Analysis.— Begin  at  the  right.  8x7'  =  56" 
=  4'  8".  Write  the  8"  one  place  to  the  right, 
reserving  the  4'  to  add  to  the  next  product. 

Then  9  ft.  x  7'  =  63'  ;   63'  +  4'  =  67'  =  5  ft.  7', 

which  write  in  the  places  of  feet  and  primes. 

Next  multiply  by  4  feet ;   8'  x  4  ft.  =  32'  = 

a  *,       ^     ^7/      2  ft.  8'.     Write  the  8'  in  the  place  of  primes, 

reserving  the  2  ft.  to  add  to  the  next  product. 

Then  9  ft.  x  4  ft.  =  36  ft.  ;  36  ft.  +2  ft.  =  38  ft,  which  write  in  the 

place  of  feet.    Adding  the  partial  products,  the  sum  equals  44  ft. 

3'  8  ",  the  product  required. 

2.  How  many  square  feet  in  4  boards,  each  12  ft.  9' 
long,  and  1  ft.  4'  wide  ? 

Kule. — I.  Write  the  terms  of  the  multiplier  under  the 
corresponding  terms  of  the  multiplicand. 

II.  Multiply  each  term  of  the  multiplicand  by  each  term 
of  the  multiplier,  beginning  with  the  lowest  order  of  units 
in  each.  Reduce  each  product  to  higher  denominations 
when  possible,  and  write  in  their  proper  places.  TJie  sum 
of  the  partial  products  ivill  be  the  product  required, 

3.  Multiply  10  ft.  6'  4"  by  5  ft.  3'  8". 

4.  Find  the  area  of  a  floor  14  ft.  8'  wide  and  16  ft.  5'  long. 

5.  What  are  the  solid  contents  of  a  block  of  marble 
6  ft.  10'  long,  4  ft.  3'  wide,  and  1  ft.  9'  thick? 


240 


DENOMINATE     NUMBERS. 


457. 


SYNOPSIS    FOE    REVIEW. 


1.  Definition  of  Reduction. 


2.  Reduction    to    Lower 
Denominations. 


is 


Principle. 
Rule,  I,  II 


)1.  Principle. 
2. 


Q 

H 
5 

O 

D 


P3 

pq 
to 

H 

i— i 
O 

P 


3.  Reduction   to   Higher 

Denominations.  (  2.  Rule,  I,  IL 

f  430.  Rule. 

432.  Rule,  I,  II.  III. 

434.  Rule. 

436.  Rule,  I,  II,  (2.) 


4.     Reduction 
of  Denominate 
Fractions. 


4.  ADDITION    OF    COMP.    NUMBERS. 

5.  SUBTRACTION 

6.  MULTIPLICATION  " 

7.  DIVISION 


.    _  (  1.  Longitude. 

1.  DEFINITIONS.    |  j^j 


r  1.   HOW    PER- 
FORMED. 

2.  Upon  what 
Principles 
based. 


H     . 

Q  W 

p  a 

£S 

o      i 

£  g 

t-5  -fll 

00 

Meridian. 
2.  Comparison  of  Longitude  and  Time. 


3.  Rules. 


1.  To  find  diff.  of  long,  when  diff. 

of  time  is  given. 

2.  To  find  diff.  of  time  when  diff. 

of  long,  is  given. 


1.  Definition. 

2.  Unit  of  Measure. 

3.  Table. 

4.  Addition  and  Subtraction. 

!1.  Product  of  two  dimensions. 
2.  Product  of  three  dimensions 
3.  Rule,  I,  II. 


4:58,  Measurements  involve  a  practical  applica- 
tion of  the  Weights  and  Measures  to  various  operations 
required  in  the  mechanic  arts,  and  to  the  common  busi- 
ness of  life. 


KECTANGULAK    SURFACES.* 


459.  A  Rectangle  is  a  plane  fig- 
ure bounded  by  four  sides,  having  all 
its  angles  right  angles. 

It  has  two  dimensions — length  and 
breadth. 


Rectangle. 


When  all  its  sides  are  equal,  it  is  called  a  Square. 

460.  The  Area  of  a  rectangle  is  the  surface  included 
within  the  lines  which  bound  it,  and  is  expressed  by  the 
number  of  times  it  contains  a  given  unit  of  measure. 

461.  The  Unit  of  Measure  for  surfaces  is  a  square 
each  side  of  which  is  a  unit  of  some  known  length. 

Thus,  the  unit  of  square  inches  is  1  square  inch  ;  of  square  f eet„ 
1  square  foot ;  of  square  yards,  1  square  yard,  etc. 

*  Measurements  of  plane  figures  requiring  a  knowledge  of  Involution  and  Evo- 
lution are  treated  at  the  close  of  this  book  under  the  head  of  "  Mensuration." 

11 


242 


DENOMINATE     NUMBERS 


The  diagram  represents  a  square  yard,  each  side  of  which  is  1  yd. 

or  3  ft.  long,  and  the  whole  is 
divided  into  square  feet,  1  sq.  ft. 
being  the  Unit  of  Measure.  In 
one  row  there  are  3  sq.  ft.,  in 
3  rows  there  are  3  times  3  sq.  ft., 
or  9  sq.  ft.  Hence  the  area  of 
1  sq.  yd.  is  9  sq.  ft. 

So  the  area  of  a  rectangle 
formed  by  2  adjacent  rows,  is  ex- 
pressed by  3  sq.  ft.  x  2,  or  G  sq.  ft. 


Square 
Foot. 

Square 
Foot. 

a  sq.  It. 


y  sq.  ft. 


462.  To  find  the  area  of  a  rectangle  : 

Eule. — Find  the  product  of  the  numbers  denoting  the 
length  and  breadth,  expressed  in  the  same  denomination  ; 
the  result  is  the  area. 

463.  To  find  either  dimension  of  a  rectangle  : 
Eule. — Divide  the  area  by  one  dimension  ;  the  quotient 

is  the  other. 

464.  Artificers  compute  their  work  by  linear y 
superficial  or  square,  and  cubic  measures. 

1.  Glazing  and  stone-cutting  are  estimated  by  the  square  foot. 

2.  Plastering,  paving,  ceiling,  etc.,  commonly  by  the  square  foot, 
or  the  square  yard. 

3.  Roofing,  flooring,  partitioning,  slating,  etc.,  generally  by  the 
square  of  100  square  feet ;  sometimes  by  the  square  foot,  or  square 
yard. 

4    One  thousand  shingles,  averaging  4  in.  wide,  and  laid  5  in. 
to  the  weather,  are  estimated  to  cover  a  square. 
5.  Bricklaying  is  generally  estimated  by  the  thousand  bricks. 

WRITTEN    EX  EJtC  I  S  ES. 

465.  1.  How  many  square  feet  in  a  floor  27  ft.  long 
and  21  ft.  wide  ?     How  many  square  yards  ? 

2.  How  many  feet  wide  is  a  hall  that  is  26  ft.  long  and 
contains  195  square  feet  ? 


MEASUKEMEKTS.  243 

3.  What  is  the  length  of  a  lawn  that  contains  305  sq.  yd. 
and  is  45  ft.  wide  ? 

Find  the  area  of  rectangles  having  the  following 
dimensions  : 

4.  12-J-  yards  square.  7.  5  ch.  14  1.  by  6  ch.  25  1. 

5.  18  yd.  2  ft.  square.       8.  25  ft.  6  in.  by  16  ft.  9  in. 
G.  18£  rd.  by  20-J  rd.        9.  14  yd.  1  ft.  10  in.  square. 

The  area  and  one  dimension  given,  find  the  other 
dimension  of  the  following  rectangles  : 

10.  Area  374£  square  feet,  length  20  ft.  6  in. 

11.  Area  5  A.  41  P.,  width  7  chains  25  links. 

12.  Area  180  sq.  yd.  4  sq.  ft.,  width  9  yd.  2  ft. 

13.  How  many  square  yards  in  the  sides  of  a  room 
16  ft.  long,  12  ft.  6  in.  wide,  and  9  ft.  3  in.  high  ? 

14.  How  many  planks  12  ft.  long  and  10  in.  wide  will 
be  required  to  floor  a  room  which  is  24  ft.  by  20  ft.  ? 

Find  the  number  of  yards  in  length  of  carpeting  re- 
quired for  rooms  of  the  following  dimensions  : 

15.  For  a  room  24  ft.  by  16  ft.  6  in.;  carpet  1  yd.  wide. 

16.  For  a  room  52  ft.  by  35  ft. ;  carpet  2  ft.  4  in.  wide. 

17.  For  a  room  28  ft.  by  23  ft.  9  in. ;  carpet  30  in.  wide. 

18.  For  a  room  27  ft.  3  in.  by  22  ft.  6  in.  ;  carpet  2  ft. 
6  in.  wide. 

Find  the  cost  of  carpeting  rooms,  their  dimensions, 
and  the  width  and  price  of  carpet  being  as  follows  : 

19.  Floor,  34  ft.  by  18  ft.  6  in.,  carpet  2  ft.  wide,  at 
$.  94  a  yard. 

20.  Floor,  30  ft.  3  in.  by  22  ft.,  carpet  f  yd.  wide,  at 
$1.08  a  yard. 

21.  Floor,  18J  ft.  by  16.4  ft.,  carpet  |  yd.  wide,  at 
$2£  a  yard. 


244  DENOMINATE     NUMBEKS. 

22.  Floor,  40  ft.  by  36  ft.,  covered  with  matting  4  ft. 
wide,  at  $1.22  a  yard. 

23.  Floor,  2G  ft.  6  in.  by  18  ft.,  covered  with  oil-cloth, 
at  $1.15  a  square  yard. 

24.  How  many  tiles  8  inches  square,  will  lay  a  floor 
48  ft.  by  10  ft.  ? 

25.  What  will  be  the  cost  of  flagging  a  side-walk  312  ft. 
long  and  6-J-  ft.  wide,  at  $2.70  a  square  yard  ? 

26.  What  will  it  cost  to  cement  a  cellar  bottom  48  ft. 
6  in.  long  and  27  ft.  wide,  at  $.45  a  square  yard  ? 

27.  How  many  squares  are  there  in  a  partition  104  ft. 

9  in.  long,  and  20  ft.  4  in.  high  ? 

28.  What  is  the  expense  of  plastering  the  sides  and 
ceiling  of  a  room  40  ft.  long,  36£  ft.  wide,  and  22J  ft. 
high,  at  $.36  a  square  yard,  allowing  1375  sq.  ft.  for  doors, 
windows,  and  baseboard  ? 

29.  Find  the  cost  of  glazing  6  windows,  each  8  ft.  3', 
by  5  ft.  4',  at  $.75  a  square  foot. 

30.  A  room  is  24  ft.  by  16J-  ft,  and  18  ft.  high.  Find 
the  cost  of  papering  its  sides  with  paper  40  in.  wide  and 
8  yd.  in  a  roll,  at  $1.20  a  roll  put  on,  and  edging  it  with 
gilt  moulding  next  the  ceiling,  at  9  cents  a  foot.  There 
are  two  windows,  each  2  ft.  4  in.  by  5  ft.  8  in.,  and  2  doors 
2  ft.  9  in.  by  6  ft.  6  in.,  and  a  baseboard  9  in.  wide. 

31.  How  many  sods,  each  16  in.  square,  will  be  required 
to  turf  a  yard  53  ft.  4  in.  long  and  28  ft.  wide  ? 

32.  How  many  yards  of  silk,  £  yd.  wide,  will  be  re- 
quired to  line  24  yd.  of  satin  j-  yd.  wide  ? 

33.  How  many  rolls  of  paper,  each  8  yd.  long  and  18  in. 
wide,  will  paper  the  sides  of  a  room  16  ft.  by  14  ft.  and 

10  ft.  high,  deducting  124  sq.  ft.  for  doors  and  windows  ? 


MEASUREMENTS.  245 

34.  Find  the  cost  of  lining  a  tank  5  ft.  8  in.  long,  4  ft. 
wide,  and  5  ft.  deep,  with  zinc,  weighing  5  lb.  to  the 
square  foot,  at  12  cents  a  pound,  which  includes  the  labor. 

35.  Find  the  cost  of  plastering  the  walls  of  a  room 
12  ft.  11'  square,  9  ft.  3'  high,  allowing  for  2  windows  and 
1  door,  each  6  ft.  2'  by  2  ft.  4',  at  £.28  a  square  yard. 

36.  How  many  shingles  4  in.  wide,  laid  6  in.  to  the 
weather,  will  cover  the  roof  of  a  building,  the  ridge  being 
4G  ft.  long,  and  the  girt  from  eaves  to  eaves  40  ft.,  the 
first  course  on  each  of  the  eaves  being  double  ? 

37.  What  will  be  the  cost  of  wainscoting  a  room  21  ft. 
8  in.  by  14  ft.  10  in.  and  10  ft.  6  in.  high,  at  $.30  a  sq.  yd.? 

38.  Find  the  cost  of  slating  a  roof  64  ft.  9  in.  long  and 
45  ft.  wide,  at  $15.3 7£  per  square  ? 

LAND. 

466.  The  Unit  of  land  measure  is  the  acre. 

Measurements  of  land  are  commonly  recorded  in  square  miles, 
acres,  and  hundredths  of  an  acre.  The  denomination  rood  is  no 
longer  used.     See  Arts.  341  and  346. 

WRITTEN     EXERCISES. 

467.  1.  How  many  acres  in  a  farm  120  rods  square  ? 

2.  A  field  80  rd.  long  contains  16  A.;  what  is  its  width  ? 

3.  A  town  6£  mi.  long  and  5£  mi.  wide  is  equal  to  how 
many  farms  of  120  A.  each  ? 

4.  What  decimal  part  of  an  acre  is  a  piece  of  land  121 
yd.  long  and  75  feet  wide  ? 

5.  A  rectangular  farm  containing  435  A.  96  P.  is  264 
rd.  long  on  one  side :  what  is  the  length  of  the  other  side? 

6.  What  is  the  value  of  a  farm  189.5  rd.  long  and  150 
rd.  wide,  at  $42f  an  acre  ? 


246  DENOMINATE     NUMBERS. 

7.  A  man  having  a  field  70  rd.  square  appropriated  5  A. 
of  it  to  corn,  100  sq.  rd.  to  garden  vegetables,  and  the 
remainder  to  meadow.  What  fractional  part  of  the  whole 
field  did  the  meadow  comprise  ? 

8.  A  rectangular  field  50  rd.  long  contains  10  acres. 
Another  field  of  the  same  width  contains  5  acres  ;  what 
is  its  length  ? 

9.  At  $2.75  a  rod,  how  much  less  will  it  cost  to  fence 
a  piece  of  land  80  rd.  square  than  if  the  same  were  in 
the  form  of  a  rectangle  twice  as  long  and  one-half  as  wide  ? 

10.  I  bought  a  piece  of  land  16  ch.  long  and  15  ch. 
wide,  at  $100  an  acre,  and  dividing  it  into  lots  of  6  rd. 
by  5  rd.,  sold  them  at  $50  each.     What  was  my  gain? 

468.  Government  Lands  are  usually  surveyed 
into  rectangular  tracts,  bounded  by  lines  conforming  to 
the  cardinal  points  of  the  compass. 

A  Base-line  on  a  parallel  of  latitude,  and  a  Principal 
Meridian  intersecting  it,  are  first  established.  Other 
lines  are  then  run  six  miles  apart,  each  way,  as  nearly  as 
possible. 

The  tracts  thus  formed  are  called  Townships,  and  con- 
tain, as  near  as  may  be,  23040  acres. 

A  line  of  townships  extending  north  and  south  is 
called  a  Range. 

The  ranges  are  designated  by  their  number  east  or  west  of  the 
'principal  meridian. 

The  townships  in  each  range  are  designated  by  their  number 
north  or  south  of  the  base-line. 

Since  the  earth's  surface  is  convex,  the  principal  meridians  con- 
verge as  they  proceed  northward.  This  tends  to  throw  the  town- 
ships and  sections  out  of  square,  and  necessitates  occasional  lines 
of  offset,  called  "  correction  lines." 


MEASUEEMENTS. 


247 


Townships  are  subdivided  into  Sections,  and  sections 
into  Half- Sections,  Quarter- Sections,  Half -Quarter- Sec- 
tions, Quarter- Quarter- Sections,  and  Lots. 

Diagram  No.  1  shows  the  sub-divisions  of  a  Tovmsldp  into  Sec- 
tions, and  how  they  are  numbered,  commencing  at  the  N.E.  corner. 

Diagram  No.  2  shows  the  sub-divisions  of  a  Section,  on  an  en 
larged  scale,  and  how  they  are  named. 


W 


Diaqram  3To.  1, 

A  Township. 
N 


6 

5 

4 

3 

2 

1 

7 

8 

9 

10 

11 

12 

18 

17 

16 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

Diagram  No*  £. 


A  Section 

N 


E     W 


N.W.  X 

of 
N.W.  H 
40  A. 


S.  W.X 

of 
N.  W.  M 


E.H 

of 

N.W.M 

80  A. 


N.  E.X 

160  A. 


320  A. 


Table. 


6mi 

x  6  mi 

.=36  sq. 

mi 

.=23040  Acres 

=  1  Township. 

1  " 

x  1   " 

=  1 

=     640 

<( 

=  1  Section. 

1  " 

x  t  " 

==  * 

=     320 

t< 

=  1  Half-Section. 

i  " 

kl  " 

—  i 

—  t 

=     160 

<< 

=  1  Quarter- Section. 

*  " 

xi    " 

—  i       < 

—  8 

=      80 

11 

=  1  Half-Quarter-Section. 

i  " 

X*    " 

=  A        ' 

=      40 

<< 

=  1  Quarter-Quarter-Section. 

A  Lot  is  a  sub-division  of  a  section,  usually  of  irregular  form,  on 
account  of  bordering  upon  a  navigable  river  or  lake — containing  as 
near  as  may  be  the  area  of  a  Quarter-Quarter-Section,  and  described 
as  lot  No.  1,  2,  3,  etc.,  of  a  particular  section. 

City  and  village  plats  are  usually  sub-divided  into  Blocks,  and 
these  into  Lots. 


248  DE^OMIKATE     K  UMBERS. 


WRITTEN    EXEItC  IS  ES. 

1.  If  a  township  of  land  is  equally  divided  among  288 
families,  how  many  acres  does  each  receive  ?  What  part 
of  a  section  ? 

2.  What  number  of  rails  will  enclose  a  quarter  section 
of  land  with  a  fence  6  rails  high,  and  3  lengths  for  every 
2  rods  ;  and  what  will  be  the  cost  of  the  rails,  at  $40  per 
thousand  ? 

3.  A  man  bought  the  S.  \  of  a  section  of  land  at  $2J- 
an  acre,  and  afterward  sold  the  E.  \  of  what  he  bought 
at  $4.3  7-§-  an  acre.     What  was  his  gain  ? 

4.  If  I  buy  the  N.E.  \  and  the  E.  \  of  N.W.  J  of  a 
section  of  land,  how  many  acres  do  I  purchase  ?  What 
part  of  a  whole  section  ?  How  are  the  parts  located  in 
respect  to  each  other  ? 

5.  A  speculator  bought  of  the  111.  Central  R.  R.  Co., 
the  S.  \  of  Section  4,  township  10  north,  range  6  east,  at 
$2  an  acre.  He  afterward  sold  the  E.  \  of  S.E.  \  at  $2.75 
an  acre ;  the  N.W.  \  of  S.E.  \  at  %Z\  an  acre ;  and  the 
N.  \  of  S.W.  \  at  $3.84  an  acre.  How  many  acres  has 
he  left  ?  What  was  his  gain  on  the  purchase  price  of 
the  whole  ?     Draw  diagram. 

6.  A  man  having  purchased  a  section  of  land  from 
the  U.  S.  Government  at  $1.25  an  acre,  sold  the  S.  \  of 
S.W.  i  at  $2.50  an  acre  ;  the  N.E.  }  of  N.W.  J  at  $1.75 
an  acre  ;  the  W.  £  of  S.  E.  \  at  $2  an  acre ;  and  the  W.  \ 
of  S.W.  i  of  N.E.  £  at  $3  an  acre.  How  many  acres 
has  he  remaining,  and  what  is  his  gain,  provided  the 
remainder  is  sold  at  $2£  an  acre  ?  Draw  diagram  and 
explain. 


MEASUREMENTS 


240 


EEOTANGULAE    SOLIDS. 

469.  A  Rectangular  Solid  is  a  body  bounded  by 
six  rectangular  plane  faces. 

The  opposite  sides  are  equal  and 
parallel. 

It  lias  three  dimensions — length, 
breadth,  and  thickness. 

When  all  its  faces  are  equal,  it  is 
called  a  Cube. 

470.  The  Volume  or  Solid  Contents  of  a  body 
is  tbe  space  included  within  the  surfaces  which  bound  it, 
and  is  expressed  by  the  number  of  times  it  contains  a 
given  unit  of  measure. 

471.  The  Unit  of  Measure  for  solids  is  a  cube,  the 
edge  of  which  is  a  unit  of  some  known  length. 

Thus,  the  unit  of  cubic  inches  is  a  cube  the  edge  of  which  is 
1  inch,  or  1  cubic  inch;  of  cubic  feet,  1  cubic  foot,  etc. 

The  diagram  repre-  __ 

sents  a  cubic  yard, 
each  face  being  a 
square  yard,  contain- 
ing 9  sq.  ft.  If  a 
section  1  ft.  thick  is 
cut  from  one  side,  it 
may  be  divided  into 
3  times  3  cu.  ft. ,  or  9 
cu.  ft.,  1  cu.  ft.  being 
the  unit  of  measure. 
And  since  the  cubic 
yard  is  3  ft.  thick,  it 
contains  3  such  sections,  or  3  times  9  cu.  ft.,  which  are  27  cu.  ft. 
Hence  the  volume  of  1  cu.  yd.  is  27  cu.  ft. 

So  the  volume  of  a  solid,  formed  of  two  adjacent  sections,  is  ex- 
pressed by  3  cu  ft.  x  3  x  2  =  18  cu.  ft. 


9  cu.  ft.  x  3  =  27  cu.  ft. 


250  DENOMINATE     NUMBERS. 

472.  To  find  the  volume  of  a  rectangular  solid  : 

Rule. — Find  the  product  of  the  numbers  denoting  the 
three  dimensions,  expressed  in  the  same  denomination; 
this  result  is  the  volume. 

473.  To  find  a  required  dimension  of  a  rectangular 
solid  : 

Rule. — Divide  the  volume  by  the  product  of  the  num~ 
bers  denoting  the  other  two  dimensions;  the  quotient  will  be 
the  required  dimension. 

WRITTEN     EXERCISES. 

474.  1.  What  are  the  contents  of  a  rectangular  solid 
6  ft.  long  and  4  ft.  square  ? 

2.  What  is  the  volume  of  a  solid  9  ft.  long,  4  ft.  wide, 
and  3  ft.  thick  ? 

3.  A  vat  12  ft.  square  contains  1224  cu.  ft.  What  is 
its  depth  ? 

4.  What  is  the  volume  of  a  bin,  the  inside  dimensions 
of  which  are  8  ft.  6  in.  by  6  ft.  by  4  ft.  4  in. 

5.  How  many  cubic  yards  of  earth  must  be  removed  in 
digging  a  cellar  36  ft.  long,  24  ft.  wide,  and  G£  ft.  deep  ? 

Find  the  volume  of  rectangular  solids  having  the  fol- 
lowing dimensions  : 

6.  Of  a  cube  the  edge  of  which  is  1  yd.  1  ft.  9  in. 

7.  Of  a  solid  6  yd.  2  ft.  7  in.  by  3  ft.  4  in.  by  2  ft.  11  in. 

8.  Of  a  solid  5  ft.  square  and  the  height  6.4  ft. 

Find  the  required  dimension  of  rectangular  solids, 
the  volumes  and  two  dimensions  being  as  follows  : 
9.  Volume,  6  cu.  ft. ;  length,  8  ft.  ;  width,  8  ft. 
10.  Volume,  20  cu.  ft;  length,  36  ft.;  width,  10  in. 


MEASUREMENTS 


251 


11.  Volume,  13  cu.  yd.  14  cu.  ft.  900  cu.  in.;  width, 
7  ft.  3  in. ;  height,  5  ft.  6  in. 

12.  How  many  cubic  feet  of  air  in  a  room  that  is  24  ft, 
9  in.  long,  18  ft.  4  in.  wide,  and  10  ft.  8  in.  liiglPS* 


A  Cord  of  wood  is 
a  pile  8  ft.  long,  4  ft.      ^j 
wide,  and  4  ft.  high.     BljjB 


A  cord-foot  is  1  foot 
in  length  of  such  a 
pile ;  that  is,  1  ft. 
long,  4  ft.  wide,  and 
4  ft.  high. 


13.  How  many  cords  in  a  pile  of  wood  30  ft.  long,  8  ft. 
wide,  and  6  ft.  6  in.  high  ? 

14.  A  pile  of  wood  containing  07J  cords,  is  90  ft.  long 
and  12  ft.  wide.     How  high  is  it  ? 

15.  What  will  be  the  cost  of  a  pile  of  wood  12  ft.  6  in. 
long,  8  ft.  wide,  and  4  ft.  6  in.  high,  at  $3.75  a  cord? 

16.  What  will  it  cost  to  dig  a  cellar  45  ft.  long,  28  ft. 
wide,  and  8  ft.  6  in.  deep,  at  $.42  a  cubic  yard  ? 

17.  What  must  be  the  length  of  a  load  of  wood  that  is 
3  ft.  high  and  5  ft.  4  in.  wide,  to  contain  a  cord  ? 

18.  How  many  cans,  8  in.  by  6  in.  by  3  in.,  can  be 
packed  in  a  box  32  in.  by  24  in.  by  15  in.  in  the  clear? 

19.  At  $3 \  a  cord,  what  is  the  value  of  the  wood  that 
can  be  piled  under  a  shed  50  ft.  long,  25  ft.  wide,  and 
12  ft.  high  ? 

20.  In  building  a  house,  200  joists  10  in.  by  3  in.  were 
used,  which  together  amounted  to  1000  cu.  ft.  What  was 
the  lcno-th  of  each  ? 


252  DENOMINATE     NUMBERS. 

475.  Masonry  is  estimated  by  the  cubic  foot,  and  by 
the  perch;  also  by  the  square  foot  and  the  square  yard. 

1.  Materials  are  usually  estimated  by  cubic  measure ;  the  work 
by  cubic  or  square  measure. 

2.  A  Perch  of  stone,  or  of  masonry,  is  16|  ft.  long,  1\  ft.  wide, 
and  1  ft.  high,  and  is  equal  to  24.75  cu.  ft. 

3.  When  stone  is  built  into  a  wall,  22  cu.  ft.  make  a  perch,  2£  cu. 
ft.  being  allowed  for  mortar  and  filling. 

4.  Embankments  and  Excavations  are  estimated  by  the  cubic  yard. 

5.  A  cubic  yard  of  common  earth  is  called  a  load. 

6.  Brickwork  is  generally  estimated  by  the  thousand  bricks;  some- 
times in  cubic  feet.  In  walls,  brick- work  is  estimated  at  the  rate  of 
a  brick  and  a  half  thick. 

7.  North  River  bricks  are  8  in.  x  3|  x  2^  ;  Maine  bricks  are  7|  in. 
x  3f  x  2 1 ;  Philadelphia  and  Baltimore  bricks  are  8^  in.  x  4 \  x  2f  ; 
and  Milwaukee  bricks  8|  in.  x  4|  x  2f . 

8.  In  estimating  material,  allowance  is  made  for  doors,  windows, 
and  cornices. 

9.  In  estimating  the  work,  masons  measure  each  wall  on  the 
outside,  and  ordinarily,  no  allowance  is  made  for  doors,  windows, 
and  corners  ;  but  sometimes  an  allowance  of  one-half  is  made,  this 
being,  however,  a  matter  of  contract. 

476.  To  find  the  number  of  bricks  in  a  cubic  foot  of 
masonry : 

Rule. — I.  Add  to  the  face  dimensions  of  the  kind  of 
bricks  used  the  thickness  of  the  mortar  or  cement  in  luhich 
they  are  laid,  and  compute  the  area. 

II.  Multiply  this  area  by  the  quotient  of  the  thickness 
of  the  ivall  divided  by  the  number  of  bricks  of  which  it  is 
composed,  the  product  will  be  the  volume  of  a  brick  and 
its  mortar  in  cubic  inches. 

III.  Divide  1728  by  this  volume,  and  the  quotient  will 
be  the  number  of  bricks  in  a  cubic  foot 


MEASUREMENTS.  253 

WRITTEN    EXERCISES. 

4I77.  1.  How  many  Milwaukee  bricks  in  a  cubic  foot 
of  wall  12f  in.  wide,  laid  in  courses  of  mortar  £  of  an  inch 
thick  ? 

OPERATION. 

8.5  +  .25  ss  8.75  in.  =  length  of  brick  and  joint. 
2.375  +  .25  =  2.625  in.  =  thickness  of  brick  and  joint. 
8.75  x  2.625  =  22.96875  sq.  in.  =  area  of  its  face. 
1275  _i_  3  (number  of  bricks  in  width  of  wall)  =  4.25  in.  =  width 
of  brick  and  mortar. 
22.96875  x  4.25  —  97.617+  =  cubic  inches  in  a  brick. 
1728  ■*?  97.617+  =  17.7+  =  number  of  bricks  in  a  cubic  foot. 

2.  How  many  bricks,  8  in.  x  4  x  2,  will  be  required  to 
build  a  wall  42  ft.  long,  24  ft.  high,  and  16 J  in.  thick, 
laid  in  courses  of  mortar  J  of  an  inch  thick  ? 

3.  How  many  perches  of  stone,  laid  dry,  will  build  a 
wall  60  ft.  long,  16 J  ft.  high,  and  18  in.  thick  ? 

Rules. — 1.  Multiply  the  number  of  cubic  feet  in  the' 
wall,  or  work  to  be  done,  by  the  number  of  bricks  in  a  cubic 
foot ;  the  product  will  be  the  number  of  bricks  required. 

2.  Divide  the  number  of  cubic  feet  in  the  icork  to.  be  done 
by  24.75  ;  the  quotient  will  be  the  number  of  perches. 

4.  How  many  perches  of  masonry  in  a  wall  120  ft.  long, 
6  ft.  9  in.  high,  and  18  in.  thick  ? 

5.  How  many  bricks,  8  J  in.  x  4  x  2J,  will  be  required 
to  build  the  four  walls  of  a  square  house  36  ft.  long,  24  ft. 
high,  and  12f  in.  thick,  allowing  224  cu.  ft.  for  doors  and 
windows,  and  £  of  an  inch  for  each  course  of  mortar  ? 

6.  At  $.56  a  cu.  yd.,  what  will  it  cost  to  remove  an 
embankment  240  ft.  long,  38  ft.  wide,  and  8.5  ft.  high  ? 


254  DENOMINATE     NUMBEES. 

7.  Find  the  cost  of  digging  and  walling  the  cellar  of  a  ■ 
house,  whose  length  is  41  ft.  3  in.,  and  width  33  ft. ;  the 
cellar  to  be  8  ft.  deep,  and  the  wall  1£  ft.  thick.     The 
excavating  will  cost  $.50  a  load,  and  the  stone  and  mason   >A4**j 
work  $3. 75  a  perch. 

8.  How  many  perches  of  stone  will  be  required  to  en- 
close a  lot  16  rd.  lonsr  and  12  rd.  wide,  with  a  wall  6  ft. 

to 

high  and  3  ft.  thick,  allowing  one-half  for  the  corners  ? 

9.  A  street  650  ft.  long  and  72  ft.  wide,  averages  4.5  ft. 
below  grade.    Find  the  cost  of  filling  it  in,  at  $.  42  a  cu.  yd.  ? 

10.  What  will  be  the  cost  of  building  a  wall  60  ft. 
long,  21  i  ft.  high,  and  17  in.  thick,  of  Philadelphia  bricks, 
laid  in  courses  of  mortar  J  of  an  inch  thick,  at  $12-}-  per  M.? 

11.  How  many  cubic  feet  of  masonry  in  the  wall  of  a 
cellar  37£  ft.  long,  26  ft.  wide,  and  9  ft.  deep,  the  wall-  /|><- 
being  2  ft.  thick,  allowing  one-half  for  the  corners  ;  and   ■  iy,i 
what  will  be  the  cost,  at  $3.85  a  perch  ?  eta  I  y^7Vx ; 

/L-tinVf'       BOARDS 'AND    TIMBER.  ****.**; 

/,  2>  l¥  r  *  Wtfff <**#"* 

478.  A  board  Foot  is  1  ft.  long;  1  ft.  wide,  and 

1  inch  thick.     Hence  12  board  feet  make  1  cubic  foot. 

Board  feet  are  changed  to  cu.  ft.  by  dividing  by  12,  and 
cubic  feet  are  changed  to  board  feet  by  multiplying  by  12. 

1.  In  Board  Measure  all  boards  are  assumed  to  be  1  in.  thick. 

2.  Lumber  and  Sawed  Timber,  such  as  plank,  scantling,  joists, 
etc. ,  are  usually  estimated  in  board  measure. 

3.  Hewn  and  Round  Timber  are  commonly  estimated  in  cubit 
measure. 

479.  When  lumber  is  not  more  than  1  inch  thick  : 
Rule. — Multiply  the  length  in  feet  by  the  width  in 

inches,  and  divide  the  product  by  12. 


MEASUBEMEKTS.  255 

480.  When  more  than  1  inch  thick  : 

Kule. — Multiply  the  length  in  feet  by  the  width  and 
thickness  in  inches,  and  divide  the  product  by  12. 

1.  If  one  of  the  dimensions  is  inches,  and  the  other  two  are  feet, 
the  product  will  be  board  feet. 

2.  If  a  board  tapers  regularly,  multiply  the  length  by  the  mean 
width,  found  by  taking  half  the  sum  of  the  two  ends. 

WRITTEN    EXERCISES, 

481.  1.  Find  the  contents  of  a  board  15  ft.  long  and 
8  in.  wide. 


Opekation .— 15  x  8  -5-  12  =  10  board  feet. 

2.  What  are  the  contents  in  board  measure  of  a  joist 
1G  ft.  long,  10  in.  wide,  and  3  in.  thick  ? 


Operation.— 3  xlOx  16-7-12  =  40  board  feet. 

3.  How  many  board  feet  in  4  boards  16  ft.  long,  10  in. 
wide  ? 

4.  How  many  board  feet  in  2  joists  17  ft.  long,  11  in. 
wide,  and  3  in.  thick  ? 

5.  Find  the  contents  of  a  board  18  ft.  long,  1  ft.  8  in. 
wide  at  one  end,  and  14  in.  at  the  other. 


Operation.— 20  in.  + 14  in.  +3=17  in. ;  18  x  17+12=251  board  ft. 

C.  Find  the  cost  of  5  boards  12  ft.  long,  17  in.  wide  at 

one  end  and  11  in.  at  the  other,  at  6  cents  a  square  foot. 

7.  Find  the  cost  of  10  planks,  each  15  ft.  long,  16  in. 
wide,  and  3£  in.  thick,  at  S3. 25  per  hundred  feet. 

8.  What  length  of  board  9  in.  wide  contains  8  board  ft.  ? 


Operation.— 144x8-4-9=128  ;  128-J-12=10f  ft.,  the  length. 
9.  What  length  may  be  cut  from  a  board  15  ft.  long 
and  20  in.  wide,  so  as  to  leave  15  board  feet  ? 


256  DENOMINATE     NUMBERS. 

10.  What  must  be  the  width  of  a  board  16  ft.  long  to 
contain  12  board  feet  ? 


Operation.— 16  ft.  =192  in. ;  144  x  12-^-192=9  in.,  the  width. 

11.  What  must  be  the  width  of  a  piece  of  board  5  ft. 
3  in.  long,  to  contain  7  square  feet  ? 

12.  Find  the  cost  of  3  pieces  of  timber,  each  26  ft.  long 
and  6  in.  by  9  in.,  at  $1.75  per  hundred  board  feet. 

13.  Find  the  cost  of  8  pieces  of  scantling,  3  in.  by  4  in. 
and  14  ft.  long,  at  $9.50  per  thousand  board  feet. 

14.  What  length  of  a  piece  of  timber  6  in.  by  9  in.,  will 
contain  3  cubic  feet  ? 


Operation.— 1728  x  3^-9  x  6=96 ;  96-^-12=8  ft.,  the  length. 

15.  A  piece  of  timber  is  10  in.  by  16  in. ;  what  length 
of  it  will  contain  15  cubic  feet  ? 

16.  What  amount  of  inch  lumber  will  make  a  box  4  ft. 
by  3  ft.  6  in.  by  2  ft.  6  in.  on  the  outside  ? 

Find  the  cost  of  the  following  :   ;-     '  ;■     \*JkyjJ$&  c 

17.  Of  36  boards,  12  ft.  long,  11  in.  wide,  ®&i  per  Or- 
,  18.  Of  16  planks,  14|  feet  long,  10  in.  wide,  and  3  in.  ~ 

thick,  @  $16£  per  M. 

19.  How  many  board  feet  in  a  stick  of  timber  36  ft. 
long,  10  in.  thick,  12  in.  wide  at  one  end  and  9  in.  wide 
at  the  other  end  ?     How  many  cubic  feet  ? 

20.  Make  a  bill  for  lumber  bought  by  John  Osborn  of 
Geo.  Mason  &  Co.,  of  St.  Paul,  Sept.  20,  1875,  as  follows  : 


124  boards, 

10  in.       by  16  ft.  @  $15      per  M. 

120       " 

16  "        "  14  "    "  S16J        " 

40  planks, 

2£xl2  "  15  «    "  $18.75      " 

96  joists, 

3x10     "  18  "    "  $14          " 

60  scantling, 

3x4       "  12  "    "  812J        " 

What  is  the  amount  ? 

MEASUREMENTS.  257 

21.  Find  the  cost  of  the  flooring  for  a  two-story  house 
at  $30  per  M.,  it  being  1£  in.  thick,  each  floor  being 
48  ft.  by  25  ft.,  no  allowance  made  for  waste. 

22.  A  rectangular  field,  16  ch.  long  and  8  ch.  wide,  is 
enclosed  by  a  post  and  board  fence  ;  the  posts  are  set  8  ft. 
apart,  the  boards  are  16  ft.  long,  and  the  fence  is  5  boards 
high.  The  bottom  board  is  12  in.  wide,  the  top  board 
6  in.,  and  the  other  three  each  9  in.  wide.  The  posts 
cost  $25  per  C,  and  the  boards  $14.80  per  M.  Required 
the  number  of  posts,  the  amount  of  lumber,  and  the  cost 
of  both.     >  UiXtT  X//£.)$7<L<*x  fcfc/M  C)  £f* 

CAPACITY    OF    BINS,    CISTERNS,    ETC. 

482.  The  Standard  Bushel  of  the  United  States 

eontains  2150.42  cu.  in.,  and  is  a  cylindrical  measure  18$ 

in.  in  diameter  and  8  in.  deep. 
• 

1 .  Measures  of  capacity  are  all  cubic  measures,  solidity  and  capaci- 
ty being  measured  by  different  units,  as  seen  in  the  tables. 

2.  The  Imperial  Bushel  of  Great  Britain  contains  2216.192  cu.  in. 

3.  The  English  Quarter  contains  8  Imp.  bushels,  or  8£  U.  S.  bu. 

4.  Grain  is  shipped  from  New  York  by  the  Quarter  of  480  lb.  (8 
tJ.  S.  bu),  or  by  the  ton  of  334-  U.  S.  bushels. 

5.  A  Register  Ton,  used  in  measuring  the  entire  internal  capacity 
or  tonnage  of  vessels,  is  100  cubic 'feet. 

6.  A  Shipping  Ton,  used  in  measuring  cargoes,  is  40  cubic  feet  in 
the  U.  S.,  and  in  England  42  cubic  feet. 

7.  The  bushel  heaped  measure  is  the  Winchester  bushel  heaped 
in  the  form  of  a  cone,  which  cone  must  be  19^  in.  in  diameter,  and 
at  least  6  in.  high. 

8.  Grain,  seeds,  and  small  fruits  are  sold  by  stricken  measure. 

9.  Corn  in  the  ear,  potatoes,  coal,  large  fruits,  coarse  vegetables, 
and  other  bulky  articles  are  sold  by  heaped  measure. 

10.  It  is  sufficiently  accurate  in  practice  to  call  5  stricken  measures 
equal  to  4  heaped  measures. 


258  DENOMINATE     NUMBEKS. 

483.  To  find  the  exact  capacity  of  a  bin  in  bushels. 
Kule. — Divide  the  contents  in  cubic  indies  by  2150.42  ; 

the  quotient  will  represent  the  number  of  bushels. 

484.  To  find  the  cubic  contents  in  a  given  number  of 
bushels. 

Kule. — 1.  Multiply  the  number  of  bushels  by  2150.42  ; 

the  product  will  be  the  number  of  cubic  inches,  which  may 

be  reduced  to  higher  denominations  if  required. 

Since  a  standard  bushel  contains  2150.42  cu.  in.,  and  a  cubic  foot 
contains  1728  cu.  in.,  a  bu.  is  to  a  cu.  ft.  nearly  as  5  to  4  ;  or  a  bu.  is 
equal  to  1£  cu.  ft.,  nearly.     Hence  for  all  practical  purposes, 

2.  Any  number  of  cubic  feet  diminished  by  -J  will  rep- 
resent an  equivalent  number  of  bushels. 

Thus,  250  cu.  ft.  -  i  of  250  cu.  ft.,  or  50  cu.  ft.  =  200,  the  num- 
ber of  bushels  in  250  cu.  ft. 

3.  Any  number  of  bushels  increased  by  J-,  will  represent 

an  equivalent  number  of  cubic  feet. 

Thus,  200  bu.  +  £  of  200  bu.,  or  50  bu.=250,  the  number  of  cubic 
feet  in  200  bushels. 

WRITTEN     EXERCISES. 

485.  1.  A  bin  is  6  ft.  long,  5  ft.  wide,  and  4  ft.  deep. 
How  many  bushels  will  it  hold  ? 

2.  A  rectangular  box  will  hold  128  bu.  What  is  its 
volume  in  cubic  feet  ?  ' 

3.  How  many  bushels  of  wheat  can  be  put  in  a  bin  8  ft. 
long,  6  ft.  6  in.  wide,  and  3  ft.  4  in.  deep  ? 

4.  What  must  be  the  depth  of  a  bin  to  contain  240  bu., 
its  length  being  10  ft.  and  its  width  5  ft.  ? 

Operation.— 240  bu.  +  60  bu.=300 ;  300-^-10x5=6  ft.,  the  depth. 

Rule. — Divide  the  contents  in  cubic  feet  or  inches  by  the 

product  of  the  two  dimensions,  in  the  same  denomination. 


MEASUREMENTS,  259 

5.  What  must  be  the  length  of  a  bin  that  is  6  ft.  wide 
and  4J-  feet  deep,  to  contain  324  bushels  ? 

6.  What  must  be  the  width  of  a  bin  12  ft.  long  and 
10  ft.  deep,  to  contain  900  bushels  of  shelled  corn  ? 

7.  A  bin  that  holds  100.8  bu.  is  7  ft.  long  and  6  ft. 
deep.     How  wide  is  it  ? 

8.  How  many  bushels  will  fill  a  bin  that  is  8.5  ft.  long, 
4.25  ft.  wide,  and  3  j  ft.  deep  ? 

9.  A  bin  10  ft.  long,  6  ft.  wide,  and  4  ft.  deep,  will  hold 
how  many  bushels  of  oats  ?     Of  potatoes  ? 

10.  How  many  bushels  of  apples  will  a  wagon-box  hold, 
that  is  12  ft.  long,  3  ft.  wide,  and  2  ft.  6  in.  deep  ?  How 
many  bushels  of  barley  ? 

11.  A  bin  20  ft.  long,  12  ft.  wide,  and  5  ft.  deep,  is  full 
of  wheat.     What  is  its  value  at  $2  a  bushel  ? 

12.  A  bin  7  ft.  long,  6  ft.  wide,  and  5  ft.  deep  is  |  full 
of  rye.     What  is  its  value  at  $1.3 7 J-  a  bushel  ? 

13.  A  farmer's  entire  crop  of  barley  just  filled  a  bin 
10  ft.  long,  6  ft.  wide,  and  5  ft.  deep.  What  was  its 
value,  at  $1.78  per  cental  ? 

14.  A  crib,  the  inside  dimensions  of  which  are  15  ft. 
long,  7  ft.  4  in.  wide,  and  8  ft.  high,  is  full  of  corn  in  the 
ear.  If  2  bu.  of  ears  make  1  bu.  of  shelled  corn,  what  is 
the  value  of  the  whole,  when  shelled,  at  $.92  a  bushel  ? 

15.  If  one  bushel  or  60  lb.  of  wheat  make  48  lb.  of 
flour,  how  many  barrels  of  flour  can  be  made  from  the 
contents  of  a  bin  10  ft.  long,  5  ft.  wide,  and  4  ft.  deep, 
filled  with  wheat  ? 

16.  How  many  tons  of  ice  can  be  packed  in  a  building 
40  ft.  long,  30  ft.  wide,  and  20  ft.  high,  a  cubic  foot  of 
ice  weighing  58-J-  pounds  ? 


260  DENOMINATE     NUMBERS. 

17.  John  Wheatley  &  Co.  bought  12400  bu.  of  wheat, 
delivered  in  New  York,  at  $1.50  a  bushel.  They  shipped 
the  same  to  Liverpool,  paying  6s.  sterling  per  quarter 
freight,  and  sold  the  entire  cargo  at  12s.  per  cental. 
Making  no  allowance  for  exchange  or  for  waste,  what 
was  the  gross  gain  in  U.  S.  Money  ? 

1.  Coal,  Ordinary  anthracite  coal  measures  from  36  to  40  cu. 
it.  to  the  ton  ;  bituminous  coal,  from  36  to  45  cu.  ft.  to  the  ton. 

2.  Lehigh,  white  ash,  egg  size,  measures  about  34|-  cu.  ft.  to  the 
ton  (2000  lb.) ;  Schuylkill,  white  ash,  35  cu.  ft.,  and  of  gray  or  red 
ash,  36  cu.  ft.  to  the  ton. 

5.  Coal  is  bought  and  sold  in  large  quantities  by  the  ton  ;  in 
small  quantities  by  the  bushel,  the  conventional  rate  being  28  bu. 
(5  pecks)  to  a  ton,  or  about  43.5  cu.  ft. 

18.  How  many  tons  of  red  ash  coal,  egg  size,  will  a 
bin  17  ft.  long,  6  ft.  wide,  and  3  ft.  deep,  contain? 

19.  A  bin  6  ft.  long,  4  ft.  deep,  and  5  ft.  9  in.  wide  is 
full  of  Lehigh  white  ash  coal.  Find  its  value  at  $6. 75  a 
ton? 

20.  A  ^arge  crib  10  yd.  long,  6  yd.  wide,  and  6  yd.  deep 
is  filled  with  Schuylkill  red  ash  coal.  Find  the  number 
of  tons  it  contains,  and  its  value  at  $5£  a  ton  ? 

21.  A  bin  7  ft.  long,  5  ft.  wide,  and  5  ft.  deep  is  half -full 
of  Schuylkill  wnite  ash  coal.  Find  its  value  at  $5.90  a  ton. 

486.  The  Standard    Liquid    Gallon  of   the 

United  States  contains  231  cu.  in.,  and  is  equal  to  about 
8-J-  lb.  Avoir,  of  pure  water. 

1.  The  half -peck,  or  dry  gallon,  contains  268.8  cubic  inches. 

2.  Six  dry  gallons  are  equal  to  nearly  seven  liquid  gallons. 

3.  The  Imperial  Gallon  of  Great  Britain  contains  277.274  cu.  in., 
and  is  equal  to  about  1.2  U.  S.  Liquid  Gallons. 


measurements.  261 

487.  Comparative  Table  of  Measures  of  Capacity. 


Cubic  in.  in 

Cubic  in.  in 

Cubic  in.  in 

Cubic  in.  La 

one  gallon. 

one  quart. 

one  pint. 

one  gill. 

iquid  Measure     .     . 

.    231 

57f 

m 

7  7 
*3T 

>ry  Measure  (<§  pk. )  . 

.    268* 

m  * 

33f 

8f 

A  cubic  foot  of  pure  water  weighs  1000  oz.,  equals  62^  lb.  Avoir. 

488.  To  find  the  exact  capacity  of  a  vessel  or  space  in 
gallons  : 

Eule. — Divide  the  contents  in  cubic  inches  by  231  for 
liquid  gallons,  or  by  268.8  for  dry  gallons. 

489.  To  reduce  gallons  to  cubic  inches  : 

Rule.— Multiply  the  given  number  of  liquid  gallons  by 
231 ;  then  reduce  to  higher  denominations  if  required. 

WRITTEN     EXERCISES. 

490.  1.  How  many  gallons  of  water  will  a  cistern 
hold,  that  is  4  ft.  square  and  6  ft.  deep  ? 

Operation.— (4  x  4  x  6  x  1728)  -*-  231  =  718^  gal. 

2.  How  many  gallons  will  a  tank  4  ft.  long,  3  ft.  wide, 
and  1  ft.  8  in.  deep  contain  ? 

3.  How  many  barrels  of  water  will  a  vat  hold  that  con- 
tains 43659  cubic  inches? 

4.  How  many  cubic  feet  in  a  space  that  holds  48  hhd.  ? 

5.  How  many  hogsheads  will  a  cistern  11  ft.  long,  6  ft 
wide,  and  7  ft.  deep  contain  ? 

6.  Find  the  weight  of  water  in  a  bath-tub  6  ft.  long, 
3  ft.  wide,  and  1  ft.  9  in.  deep. 

7.  How  many  gallons  will  a  space  contain  that  is  22.5 
ft.  long,  3.25  ft.  wide, 'and  6.4  ft.  deep? 


262  DENOMINATE      NUMBERS. 

8.  A  man  constructed  a  cistern  to  hold  32  hhd.,  the 
bottom  being  6  ft.  by  8  ft.     What  was  its  depth  ? 

9.  How  many  more  cubic  inches  in  189.5  gallons  dry 
measure  than  in  189.5  gallons  liquid  measure  ?• 

10.  Find  the  number  of  gallons  in  a  cubic  foot,  correct 
to  4  decimal  places. 

11.  A  tank  in  the  attic  of  a  house  is  6  ft.  6  in.  long, 
4  ft.  wide,  and  3  ft.  6  in.  deep.  How  many  gallons  of 
water  will  it  hold,  and  what  will  be  its  weight  ? 

12.  If  64  quarts  of  water  are  put  into  a  vessel  that  will 
exactly  hold  64  quarts  of.  wheat,  how  much  will  the  vessel 
lack  of  being  full  ? 

13.  If  a  man  buy  10  bu.  of  chestnuts  at  $5  a  bushel, 
dry  measure,  and  sell  the  same  at  25  cents  a  quart,  liquid 
measure,  how  much  does  he  gain  ? 

14.  A  cistern  5  ft.  by  4  ft.  by  3  ft.  is  full  of  water.  If 
it  is  emptied  by  a  pipe  in  1  hr.  30  min.,  how  many  gal- 
lons are  discharged  through  the  pipe  in  a  minute  ? 

15.  A  vat  that  will  hold  5000  gallons  of  water  will 
hold  how  many  bushels  of  corn  ? 

16.  A  tank  is  7  ft.  deep,  4  yd.  long, "and  3  yd.  broad. 
What  weight  of  water  is  in  it  when  just  half-full  ? 

17.  A  cellar  40  ft.  long,  20  ft.  wide,  and  8  ft.  deep  is 
half-full  of  water.  What  will  be  the  cost  of  pumping  it 
out,  at  6  cents  a  hogshead  ? 

18.  A  reservoir  24  ft.  8  in.  long  by  12  ft.  9  in.  wide  is 
full  of  water.  How  many  cubic  feet  must  be  drawn  off 
to  sink  the  surface  1  foot  ?    How  many  gallons  ? 

19.  How  many  imperial  gallons  will  a  cistern  contain 
fehat  is  7  ft.  3  in.  long,  3  ft.  8  in.  deep,  and  2  ft.  10  in.  wide? 


MEASUBEMENTS.  263 

491.  The  Avoirdupois  JPotnid  contains  7000 
Troy  grains  ;  hence  the  Troy  pound  is  fJJ-J-  =  \\%  of  an 
Avoir,  pound  ;  but  the  Troy  ounce  contains  -^ff-0-  =  480 
grains,  and  the  Avoir,  ounce,  ij-jp.  —  437.5  grains. 

The  pound,  ounce,  and  grain  of  Apothecaries'  Weight  are  the 
same  as  those  of  Troy  Weight,  the  ounce  being  differently  divided. 

493.     Comparative  Table  of  Weights. 

Troy.  Avoirdupois.  Apothecaries'. 

1  pound     =     5760  grains  =  7000  grains  =  5760  grains. 

1  ounce     =      480      "  =  437.5    "  =      480      " 

175  pounds  =  144  pounds  =      175  pounds. 

WRITTEN    EXERCISES. 

493.  1.  Change  10  lb.  8  oz.  Avoir,  weight  to  Troy. 

Operation.— 10  lb.  8  oz.  =  168  oz. ;  168  oz.  x  437.5  =  73500  gr. ; 
and  73500  gr.  h-  480  =  153^  oz.  s=  12  lb.  9^  oz.  Troy. 

2.  Change  15  lb.  10  oz.  12  pwt.  Troy  to  Avoirdupois. 

Operation.— 15  lb.  10  oz.  12  pwt.  =  190.6  oz. ;  190.6  oz.  x  480  = 
91488  gr. ;  and  91488  gr.  -r-  437.5  =  209*-? f  oz.  =  I'd  lb.  l^f  oz. 

3.  Find  the  value  in  Troy  weight,  of  9  lb.  10  oz.  Avoir. 

4.  What  is  the  value  in  Avoirdupois  weight,  of  16  lb. 
8oz.  10  pwt.  12  gr.  Troy? 

5.  What  is  the  value  of  a  coffee  urn,  weighing  2  lb. 
14  oz.  Avoir.,  at  $1.80  per  ounce  Troy  ? 

6.  How  many  ounces  of  gold  are  equal  in  weight  to 
6  lb.  of  lead  ? 

7.  If  8  lb.  Avoir,  of  drugs  are  bought  for  $12£  a 
pound,  and  retailed  at  the  rate  of  $16.25  a  pound,  Apothe- 
caries' weight,  what  is  the  gain  on  the  whole  ? 

8.  What  is  the  difference  in  the  weight  of  42f  lb.  of 
iron  and-42.375  lb.  of  gold? 


264 


DENOMINATE     NUMBERS, 


494. 


SYNOPSIS    FOE    REVIEW. 


Eh 

g 
1 

H 

Pi 

< 

9 

w 
p 

M 

o 

Q 


S3 

o 
p 


Rules. 


Rectangle. 

Area  of  Rectangle. 

Unit  of  Measure. 

1.  To  find  area  of  red. 

2.  To  find  either  dimen. 
2.  Artificers'  Work.    How  computed. 

r  Range.  "^ 

J  Township.  V  Table. 

|  Sub-divisions.    J 


Rectangular 
Surfaces. 


Government 
Lands. 


Rectangular 
Solids. 


5.  Masonry. 


G. 


Boards  and 
Timber. 


Capacity  of 
Bins,  etc. 


1.  Definitions. 


2. 


3.  Rules. 


2.  Rules. 


2.  Rules. 


1.  Beat.  Solid. 

2.  Volume. 
Unit  of  Measure. 

1.  To  find  volume  of  a 
rectangular  solid. 

2.  To  find  a  required 
dimension. 

How  estimated. 

1 .  To  fin  d  number  of  bricks 
in  a  cu.ft.  of  masonry. 

2.  To  find  number  of  Pch. 
of  stone  in  a  given  work. 

Definition  of  board  foot. 

Rules,  1,  2. 

Standard  Bushel  of  U.  S. 

1 .  To  find  capacity  of  bins. 

2.  To  find  cu.  contents  of 
a  given  number  of  gal. 

3.  To  find  either  dimen. 
Standard  Liquid  Gallon  of  U.  S. 

4.  Comp.  Table  of  Meas.  of  Capacity. 

1.  To  find  capacity  of  a 
vessel  in  gallons. 

2.  To  reduce  gal.  to  cu.  in. 

3.  To  reduce  cu.ft.  to  bu. 

4.  To  reduce  bu.  to  cu.  ft. 

5.  To  find  1  dimen.ofa  bin. 


5.  Rules.  « 


<m  PEBOinmei  m> 


Tg^  €^gjT%^  ^Njp 


^A^^ 


ORAZ     EXERCISES. 

495.  1.  What  is  T^  of  $100?    Tfe?    ^fr?    rffr? 

2.  What  is  ythj-  of  $500  ?     Of  $700  ?     Of  $1000  ? 

3.  What  is  Tfo  of  $600?    Tfc?    ffo?     £?»? 

4.  How  many  hundredths  of  $100  are  $5  ?     $7  ?    $18  ? 

5.  How  many  hundredths  of  $500  are  $25  ?  $35  ?  $50  ? 

6.  How  many  hundredths  of  any  number  is  J  of  it  ?  -£  ? 

496.  Percentage  is  a  term  applied  to  computations 

in  which  100  is  employed  as  a  fixed  measure,  or  standard. 

The  term  percentage  is  also  used  to  denote  the  result  found  by 
applying  that  standard  to  any  given  number. 

497.  -Per  Cent,  is  an  abbreviation  of  the  Latin 
phrase  per  centum,  which  signifies  by  the  hundred. 

Thus,  5  per  cent,  means  5  of  every  100,  or  T-{^,  the  5  standing  for 
the  numerator,  and  the  words  "  per  cent."  for  the  denominator  100. 

The  phrase  "per  cent.,"  wherever  it  occurs,  should  invariably 
suggest  to  the  mind  a  decimal  to  the  hundredths  place.  Thus, 
25  per  cent.  —  i2^  or  .25. 

498.  The  Sign  of  JPer  Cent,  is  %.    It  is  read  per 

cent.     Thus  6^  is  read  6  per  cent. 

6  per  cent.,  6%,  yf^and  .06  are  equivalent  expressions  ;  the  first 
two  are  used  in  the  statement  of  questions,  the  other  two  in  per- 
forming the  operations. 

7.  How  many  hundredths  of  a  number  is  11%  of  it  ?  9%? 
Si%?     15%?     8f#?     6f^?    25^?     18*#?     45#? 

12 


266 


PERCENTAGE 


8.  What  per  cent,  of  a  number  is  -^  of  it  ?  -^  ?    .08  ? 
.12*?    T%?    tW?    -025?    .00*  ?    .04|?    .375?    .0325? 

499.  What  per  cent,  of  a  number  is  £  of  it  ? 
Analysis. — Since  the  whole   of  any  number  is  £$$,  £  of  the 


SS-1- 
same  is  |  of  f{$,  or  — g,  equal  to  33-*-%, 


Hence,  etc. 


What  $  of  a  number  is  \  of  it ?    }?    |?    £?    £?    £? 

I?   V   A?   H?   1?   A?   «? 

500.  What  fractional  part  of  a  number  is  12|-$  of  it  ? 

Analysis.— 12^%  is  — r^,  or  f^,  equal  to  \.    Hence,  etc. 

What  part  of  a  number  is  8£$  of  it?    16f$?    15$? 

20$?    37£$?     7^?     G£$?    25$?     66f?     75$? 

501.  What  part  of  a  number  is  *$  of  it  ? 


Analysis.  - 


%  is  —-,  equal  to  ^.     Hence,  etc. 
1UU 


-What  is   J$  of  a  number?    ^$?    |$?    ^$?    f$? 

502.  Any  j9er  cent,  may  be  expressed  either  as  a  deci- 
mal or  as  a  fraction,  as  shown  in  the  following 

Table. 


Per  cent. 

Decimal.          Fraction. 

Per  cent. 

Decimal. 

Fraction 

1% 

•01            -rfcr- 

75$ 

.75 

i 

n 

•  02              A 

100$ 

1.00 

4$ 

•04             * 

125$ 

1.25 

H 

6$ 

•  06              A 

Wo 

.005 

7TT 

10$ 

.10,  or  .1    ^ 

i% 

.0075 

if  (7 

20$ 

.20,  or .2     \ 

m 

.08* 

A 

25$ 

.25               J 

1H% 

.125 

4 

50$ 

.50               i 

m% 

.1625 

H 

PEKCEKTAGE.  267 

WRITTEN     EXERCISES. 

503.  Change  to  expressions  having  the  per  cent.  sign. 

1.  .15;  .085;  .33f;  .375;  .00$;  lj;  1|;  .75|. 

2.  2i;  &;  .00*;  Jf;  tS  *'j  .00125  ;}  ;  2f. 
Change  to  the  form  of  decimals, 

3.  5|^;  9^;  Styf;  3^;  f£;  T« ;  lj£;  112^. 
Change  to  the  form  of  fractions, 

4.  24%';  f£;  6J# ;  37^;  |# ;  3^;  120^;  75^. 

504.  In  the  applications  of  percentage,  at  least  three 
elements  are  considered,  viz. :  the  Rate,  the  Base,  and  the 
Percentage.    Any  two  being  given,  the  other  can  be  found. 

505.  The  Mate  is  the  number  per  cent,  or  the  num- 
ber of  hundredths.     Thus,  in  5%,  .05  is  the  rate.    Hence, 

Bate  per  cent,  is  the  decimal  which  denotes  how  many  hundredths 
of  a  number  are  to  be  taken  or  expressed. 

506.  The  Base  is  the  number  of  which  the  per  cent. 
is  taken. 

Thus,  in  the  expression,  5%  of  $15,  the  base  is  $15. 

507.  The  Percentage  is  the  result  obtained  by 
taking  a  certain  per  cent,  of  the  base. 

Thus,  in  the  statement,  6%  of  $50  is  $3,  the  rate  is  .06,  the  base 
$50,  and  the  percentage  is  $3. 

508.  The  Amount  is  the  sum  of  the  base  and  the 
percentage. 

Thus,  if  the  base  is  $80,  and  the  percentage  $5,  the  amount  is 

$80  +  $5  =  $85. 

509.  The  Difference  is  the  remainder  found  by 
subtracting  the  percentage  from  the  base. 

Thus,  if  the  base  is  $80,  and  the  percentage  $5,  the  difference  is 

$80  -  $5  =  $75. 


268 


PERCENTAGE 


510.  The  base  and  rate  being  given  to  find  the 
percentage. 

O  It  A  I       EXERCISES, 


1.  What  is  10%  of  140  ? 

Analysis.— 10$  is  ffo  =  ro>  and  tV  of  140  is  14.    Hence  10% 
of  140  is  14. 


What  is 

2.  5%  of  $80? 

3.  7%  of  200  lb.  ? 

4.  6%  of  150  men? 

5.  25$  of  120  mi.? 

Find  the  amount 

10.  Of  100  A.  +  27%. 

11.  Of  $75  +  5%. 

12.  Of  32doz.  +  12}%. 


How  much  is 

6.  12£%  of  72  gal.  ? 

7.  40%  of  60  sheep  ? 

8.  8%  of  50  bu.  ? 

9.  50%  of  $240  ? 

Find  the  difference 

13.  Of  90  hhd.  —  10%, 

14.  Of  63  Od.  -  33J%. 


15.    Of  $200 


H%- 


16.  A  farmer  had  150  sheep,  and  sold  20%  of  them. 
How  many  had  he  left  ? 

17.  A  mechanic  who  received  $20  a  week  had  his  sal- 
ary increased  8%.    What  were  his  daily  wages  then  ? 

18.  From  a  hhd.  of  molasses  33£%  was  drawn.     How 
many  gallons  remained  ? 

19.  A  grocer  bought  150  dozen  eggs,  and  found  16|% 
of  them  bad  or  broken.     How  many  were  salable  ? 

20.  A  train  of  cars  running  25  miles  an  hour  increases 
its  speed  12£%.     How  far  does  it  then  run  in  an  hour  ? 

511.  Principle. — The  percentage  of  any  number  is 
the  same  part  of  that  number  as  the  given  rate  is  of  100%. 


PERCENTAGE. 


269 


WRITTEN     EXERCISES. 

512.  1.  What  is  17%  of  84957  ? 

OPERATION. 


$4957 
.17 


Analysis.— Since  17%  is  .17,  the  required 
percentage  is  .17  of  $4057,  or  $4957  x  .17,  which 
is 


$842.69 

What  is 

2.  35%  of  695  lb.  ? 

3.  75%  of  $8428  ? 

4.  12£%  of  £2105  ? 


Find 

5.  33£%  of  8736  bu. 

6.  \%  of  $35000. 

7.  120%  of  $171.24. 

Rule. — Multiply  the  base  by  the  rate.     Or,  take  such  a 
part  of  the  base  as  the  rate  is  of  100%. 

This  rule  may  be  briefly  expressed  by  the  following 
Formula. — Percentage  =  Base  x  Mate. 


What 

8.  Is  41$  of  312. 8  rd.? 

9.  Is  105%  of  $5728? 

10.  Is  $3140.75  +  1J%? 

11.  Is2|mi.+7i%? 

12.  Is400ft.-3£%? 


Find 

13.  84%  of  254  bu. 

14.  25%  of  -J  of  a  ton. 

15.  |%  of  16400  men. 

1 6.  f  %  of  |  of  a  year. 

17.  |%  of  f |  of  a  hhd. 


18.  The  bread  made  from  a  barrel  of  flour  weighs  35% 
more  than  the  flour.    What  is  the  weight  of  the  bread? 

19.  A  man  having  a  yearly  income  of  $4550  spends  20% 
of  it  the  first  year,  25%  of  it  the  second  year,  and  37£%  of 
it  the  third  year.     How  much  does  he  save  in  3  years  ? 

20.  A  man  receives  a  salary  of  $1600  a  year.  He  pays 
18%  of  it  for  board,  8£%  for  clothing,  and  16%  for  inci- 
dentals. What  are  his  yearly  expenses,  and  what  does  he 
save? 


270 


PERCENTAGE. 


21.  A  man  owning  f  of  a  cotton-mill,  sold  35%  of  his 
share  for  $24640.  What  part  of  the  whole  mill  did  he 
still  own,  and  what  was  its  value  ? 

22.  Smith  had  $5420  in  bank.  He  drew  out  15$  of  it, 
then  20%  of  the  remainder,  and  afterward  deposited  12|-% 
of  what  he  had  drawn.     How  much  had  he  then  in  bank  ? 

513.  The  base  and  percentage  being  given  to  find 
the  rate. 


orai,  exercises. 

1.  What  per  cent,  of  25  is  3  ? 

Analysis. — Since  3  is  s*f  of  25,  it  is  ^  of  100$ ,  or  12% .    Hence, 
3  is  12%  of  25. 


What  per  cent. 

2.  Of  24  is  18  ? 

3.  Of  $16  are  $4  ? 

4.  Of  200  figs  are  20  figs? 

5.  Of  40  lb.  are  15  lb.  ? 

6.  Of  12£  bu.  are  2£  bu.  ? 

7.  Of  2  A.  are  80  sq.  rd.  ? 


What  per  cent. 
9.  Are  6i  mi.  of  12-Jmi.? 

10.  Are  18  qt.  of  30  qt.  ? 

11.  Are  16|  cents  of  $1  ? 

12.  Is  $i  of  $25  ? 

13.  Is  |  of  f? 

14.  Isf  of  24? 

15.  Is  I  of  3|? 


8.  Of  1  da.  are  16  hr.  ? 

16.  I  of  an  acre  is  what  per  cent,  of  it? 

17.  I  of  a  cargo  is  what  per  cent,  of  it  ? 

18.  2J  times  a  number  is  what  per  cent,  of  it? 

19.  If  $6  are  paid  for  the  use  of  $30  for  a  year,  what  is 
the  rate  per  cent.  ? 

20.  If  a  milkman  adds  1  pint  of  water  to  every  gallon 
of  milk  he  sells,  what  per  cent,  does  he  add  ? 

514.  Principle. — The  rate  is  the  number  of  hundredths 
which  the  percentage  is  of  the  base. 


PERCENTAGE. 


271 


WRITTEN    EXERCISES. 


515.  1.-  What  per  cent,  of  72  is  48  ? 

OPERATION. 

48-^-72  =  .66f  =  m\% 
Or,  #  =  f  ;  100^  x|=  66\% 


Analysis. —Since  the  per. 
centage  is  the  product  of 
the  base  and  rate,  the  rate 
is  the  quotient  found  by  di- 
viding the  percentage  by 
the  base  ;  and  48  divided  by  72  is  f  |  =  §  =  .06}  j  hence  the  rate  is 


Since  48,  the  percentage,  is  f  of  the  base,  the  rate  is  f  of  100.%, 
or66|%. 


What  per  cent. 

5.  Of  $18  are  90  cents  ? 

6.  Of  560  lb.  are  80  lb.  ? 

7.  Of  980  mi.  are  49  mi.  ? 

Divide  the  percentage  by  the  base.     Or,  take 


What  per  cent. 

2.  Of  300  is  75? 

3.  Of  66  is  1GJ? 

4.  Of  $20  are  $21.60  ? 

Rule. 


such  apart  of  100%  as  the  percentage  is  of  the  base. 
Formula. — Rate  =  Percentage  ^  Base. 

What  per  cent. 

14.  Are 448  da. of  5600  da.? 

15.  Are  5  lb.  10  oz.  of  15  lb. 


What  per  cent. 

8.  Of  $480  are  $26.40  ? 

9.  Of  192  A.  are  120  A.  ? 

10.  Of  15  mi.  are  10.99  mi.  ? 

11.  Of  46  gal.  are  5  gal.  3  qt.? 

12.  Of  U  are  30  cents  ? 

13.  Of  6  bu.  1  pk.  are  4  bu. 

2  pk.  6  qt.  ? 

20.  A  grocer  sold  from  a  hogshead  containing  600  lb. 
of  sugar,  }  of  it  at  one  time,  and  £  of  the  remainder  at 
another  time.    What  per  cent,  of  the  whole  remained  ? 

21.  A  merchant  owes  $15120,  and  his  assets  are  $9828. 
What  per  cent,  of  his  debts  can  he  pay  ? 


Avoir.  ? 
16.  Is  13.5  of  225  ? 

17.  isfiofjyv? 

18.  Is  3|  of  18£? 

19.  Is  22|  of  182.4  ? 


272 


PERCENTAGE, 


516.  The  rate  and  percentage  being  given  to  find 
the  base. 


ORAL    EXERCISES. 

1.  18  is  3$  of  what  number  ? 

Analysis.  —Since  3%,  or  T|¥,  of  a  certain  number  is  18,  y^  is  £ 
of  18,  or  6,  and  ffg  is  600.    Hence  18  is  3%  of  600. 

Of  what  number 


2.  Is  15  25$  ? 

3.  Is  24  75$  ? 

4.  Is  48  8$? 

5.  Is  1.2  6$? 


Of  what  are 

6.  30  1b.  20$?    25$? 

7.  $84  12$?    21$? 

8.  15  bu.  30$  ?    50$  ? 

9.  16doz.  12  J$?     8£$? 


10.  12£$  of  96  is  33|$  of  what  number  ? 

517.  Principle. — The  base  is  as  many  times  the  per- 
centage as  100$  is  times  the  rate. 

WRITTEN    EXERCISES, 

518.  1.  144  is  75$  of  what  number  ? 


OPEKATTON. 

144-f-.75  =  192 


Analysis.— Since  the  percent- 
age is  the  product  of  the  base  by 
the  rate,  the  base  is  equal  to  the 
percentage  divided  by  the  rate ; 
and  144 -- .75  is  192.    Or, 

Since  the  rate  is  .75,  the  per- 
centage is  T7/7,  or  £  of  the  base  ;  hence  the  base  is  f  of  the  percent- 
age, and  f  of  144  is  192. 


Or,  100  -f-  75  =  iiyi  -  £ 
144  x  i  =  192 


2.  854  are  15$  of  what  ? 

3.  $18.75  are  %\%  of  what  ? 


4.  4.56  A.  are  5$  of  what  ? 

5.  39.6  lb.  are  7£$  of  what? 


Rule. — Divide  the  percentage  by  the  rate.     Or,  take  as 
many  times  the  percentage  as  100$  is  times  the  rate. 
Formula. — Base  =  Percentage  —  Rate. 


PERCENTAGE.  273 


Of  what  number 

Of  what 

6.     Is  828    120$? 

10.    Are  $281.25 

37£$? 

7.     Is  6119  105J$? 

11.    Are  $4578 

84$? 

8.     Is  .43      %\%  ? 

12.     Are  37£  bu. 

<*#? 

9.     Is  31*    31£$? 

13.    Are  1260  bbl. 

m%? 

14.  25$  of  800  bu.  is  %\%  of  how  many  bushels  ? 

15.  A  farmer  sold  3150  bushels  of  grain  and  had  30$ 
of  his  entire  crop  left.    What  was  his  entire  crop  ? 

16.  A  man  drew  25$  of  his  bank  deposits,  and  expended 
33j5£  of  the  money  thus  drawn  in  the  purchase  of  a  horse 
worth  $250.     How  much  money  had  he  in  bank  at  first  ? 

17.  If  a  man  owning  45$  of  a  steamboat  sells  16f  %  of 
his  share  for  $5860,  what  is  the  value  of  the  whole  boat  ? 

18.  If  $295.12  are  13£$  of  A's  money,  and  4f$  of  A's 
money  i*5  8$  of  B's,  how  much  more  money  has  A  than  B  ? 

519.  The  amount,  or  the  difference,  and  the  rate 
being  given  to  find  the  base. 

ORAL      EXERCISES. 

1.  What  number  increased  by  25$  of  itself  amounts 
to  60? 

Analysis. — Since  60  is  the  number  increased  by  25%  of  itself, 
it  is  !§§,  or  £  of  the  number  ;  and  if  £  of  the  number  is  60,  the 
number  itself  is  4  times  \  of  60,  or  48. 

2.  What  number  increased  by  8\%  of  itself  is  130  ? 

3.  $70  are  40$  more  than  what  sum  ? 

4.  A  man  sold  a  saddle  for  $18,  which  was  12J$  more 
than  it  cost  him.     What  did  it  cost  him  ? 

5.  A  grocer  sold  flour  for  $8.40  a  barrel,  which  was  16|$ 
more  than  he  paid  for  it.     What  did  he  pay  for  it  ? 


274  PERCENTAGE. 

6.  What  number  diminished  by  20$  of  itself  is  40  ? 
Analysis.— Since  40  is  the  number  diminished  by  20%  of  itself, 

it  is  -j8^,  or  f  of  the  number  ;  and  if  £  of  the  number  is  40,  the 
number  itself  is  5  times  £  of  40,  or  50. 

7.  What  number  diminished  by  5$  of  itself  is  38  ? 

8.  What  sum  diminished  by  50$  of  itself  is  $20.50  ? 

9.  68  yd.  are  15$  less  than  what  number  ? 

10.  A  tailor,  after  using  75$  of  a  piece  of  cloth,  had  9J 
yards  left.     How  many  yards  in  the  whole  piece  ? 

11.  A  sells  tea  at  $.90  a  pound,  which  is  10$  less  than 
he  paid  for  it.     What  did  he  pay  for  it  ? 

WRITTEN    EXERCISES. 

520.  1.  What  sum  increased  by  37$  of  itself  is  $2055  ? 

operation.  Analysis.— Since 

1-J-.37  =  1.37  the  number  is  in- 

$2055-^-1.37  =  $1500  creased  37^'  or  b^ 

.37  of  itself,  $2055 

0r>  is  137$,  or  1.37  the 

Hf  Of  $2055  =  $2055-^137Xl00=:$1500  number.      Hence 

A  $2055    divided    by 

1.37,  is  the  base  or  required  number.     Or, 

Since  $2055,  the  amount,  is  \ffi  of  the  base,  100  times  y|T  of 

$2055,  or  $1500,  is  the  base. 

2.  What  number  increased  by  18$  of  itself  equals  2950  ? 

3.  What  sum  increased  by  15$  of  itself  is  $6900? 

4.  What  number  diminished  by  12$  of  itself  is  2640  ? 

operation.  Analysis.— Since  the  number 

1 12  =    88  is  diminished  12%,  or  by  .12  of 

2640  -.88  =  3000  itee,f>  2640JS  W?.'  °r  f.  fAT 
number.    Hence  2640  divided  by 

Or,    2640^22  X  25  =  3000     .88  is  the  base  or  required  num- 
ber.    Or, 
Since  2640,  the  difference,  is  T%  or  f  f  of  the  base,  25  times  -fa  of 
2640,  or  3000,  is  the  base. 


PERCENTAGE.  275 

5.  If  the  difference  is  $1000  and  the  rate  20$,  what  is 
the  base  ? 

6.  What  sum  diminished  by  35$  of  itself  equals  $4810  ? 

Kule. — Divide  the  amount  by  1  plus  the  rate;   or, 
divide  the  difference  by  1  minus  the  rate. 

_    j  Amount  -^  (1  +  Rate). 
\  Difference  -f-  (1  —  Rate). 


What  number  increased 

7.  By  12$  of  itself  is  3800? 

8.  By  10$  is  39600  ? 

9.  By  15$  is  $2610.25? 
10.  By  22$  is  1098  bu.  ? 


What  number  diminished 

11.  By  7£$  of  itself  is  740? 

12.  By  4$  is  312  acres  ? 

13.  By  8$  is  $2281.60? 

14.  By  37J$  is  $234,625? 


15.  A  man  sold  160  acres  of  land  for  $4563.20,  which 
was  8$  less  than  it  cost.     What  did  it  cost  an  acre  ? 

16.  A  speculator  bought  48  bales  of  cotton,  and  after- 
ward sold  the  whole  for  $2008.80,  losing  7$.  What  was 
the  cost  of  each  bale  ? 

17.  A  dealer  bought  a  quantity  of  grain  by  measure  and 
sold  it  by  weight,  thereby  gaining  1J$  in  the  number  of 
bushels.  He  sold  at  10$  above  the  purchase  price,  and 
received  $4910.976  for  the  grain.     Required  the  cost. 

18.  A  merchant,  after  paying  60$  of  his  debts,  found 
that  $3500  would  discharge  the  remainder.  What  was 
his  whole  indebtedness  ? 

19.  The  net  profits  of  a  mill  in  two  years  were  $6970, 
and  the  profits  the  second  year  were  5$  greater  than  the 
profits  the  first  year.     What  were  the  profits  each  year? 

20.  A  man  sold  two  houses  at  $2500  each  ;  for  one  he 
received  20$  more  than  its  value  and  for  the  other  20$ 
less.     Required  his  loss. 


276  PERCENTAGE. 

APPLICATIONS    OF    PERCENTAGE. 

521,  The  applications  of  percentage  are  those  which 
are  independent  of  time,  as,  Profit  and  Loss,  Commission, 
Stocks,  etc. ;  and  those  in  which  time  is  considered,  as, 
Interest,  Discount,  Exchange,  etc. 

Since  some  one  of  the  four  formulas  of  percentage 
already  considered  will  apply  to  any  of  these  applications, 
the  following  will  serve  as  a  general 

Rule. — Note  what  elements  of  Percentage  are  given  in 
the  problem,  and  ivhat  element  is  required,  and  then  apply 
the  special  rule  or  formula  for  the  corresponding  case. 

PROFIT    AND    LOSS. 

522.  Profit  and  Loss  are  terms  used  to  express 
the  gain  or  loss  in  business  transactions. 

523*  Gains  and  losses  are  usually  estimated  at  a  rate 
per  cent,  on  the  cost,  or  the  money  or  capital  invested. 

524.  The  operations  involve  the  same  principles  as 
those  of  Percentage. 

525.  The  corresponding  terms  are  the  following  : 

1.  The  Base  is  the  Cost,  or  capital  invested. 

2.  The  Rate  is  the  per  cent,  of  profit  or  loss. 

3.  The  Percentage  is  profit  or  loss. 

4.  The  Amount  is  the  cost  plus  the  profit,  or  the 
Selling  Price. 

5.  The  Difference,  is  the  cost  minus  the  loss,  or  the 
Selling  Price. 


PROFIT     AND     LOSS.  277 

ORAL     EXERCISES, 

526.  1.  A  horse  that  cost  $200  was  sold  at  a  gain  of 

12$.     What  was  the  gain,  and  the  selling  price  ? 

Analysis. — Since  the  gain  was  12%,  it  was  Tyff  of  $200,  which  is 
$24  ;  and  the  selling  price  was  $200  +  $24 =$224.  Hence,  etc.  (510.) 

2.  A  saddle  that  cost  $25  sold  at  a  loss  of  10$.  What 
was  the  loss,  and  the  selling  price  ? 

3.  A  tailor  bought  cloth  at  $6  a  yard,  and  wished  to 
sell  it  at  a  gain  of  25$.     At  what  price  must  he  sell  it  ? 

4.  For  how  much  must  a  grocer  sell  tea  that  cost  $.60 
a  pound,  to  gain  30$  ? 

5.  A  merchant  buys  gloves  at  $.75  a  pair,  and  sells  them 
at  a  profit  of  33  £-$.     For  how  much  does  he  sell  them  ? 

6.  Bought  a  carriage  for  $160,  and,  after  paying  10$ 
for  repairs,  sold  it  at  12£$  profit.  What  was  the  gain, 
and  the  selling  price  ? 

7.  If  butter  bought  at  36  cents  a  pound  is  sold  at  a  loss 
of  16f$,  what  is  the  selling  price  ? 

8.  What  must  be  the  selling  price  of  coffee  that  cost 
25  cents  a  pound,  in  order  to  gain  20$? 

9.  At  what  price  must  an  article  that  cost  $5  be  sold, 
to  gain  100$?    120$?     150$?    200$? 

527.  1.  A  merchant  bought  cloth  at  $5  a  yard,  and 

sold  it  at  $6  a  yard.     What  was  the  gain  per  cent.  ? 

Analysis. — The  whole  gain  is  the  difference  between  $6  and  $5, 
which  is  $1.  Since  $5  gain  $1,  or  \  of  itself,  the  gain  per  cent,  is 
£of  100%  or  20%.     Hence,  etc.    (513.) 

2.  What  is  gained  per  cent,  by  selling  coal  at  $7  a  ton, 
that  cost  $6  a  ton  ? 

3.  Sold  a  piano  for  $300,  which  was  J  of  what  it  cost. 
What  was  the  loss  per  cent.  ? 


278  PERCENTAGE. 

4.  Sold  melons  for  $.75  that  cost  $.50.  What  was  the 
gain  per  cent.  ? 

5.  What  is  gained  per  cent,  by  selling  pine-apples  at  30 
cents  each,  that  cost  $15  a  hundred  ? 

6.  Sold  a  sewing  machine  at  a  loss  of  -|  of  what  it  cost. 
What  was  the  loss  per  cent.  ? 

7.  What  %  is  gained  on  goods  sold  at  double  the  cost  ? 

8.  W^hat  %  is  lost  on  goods  sold  at  one-half  the  cost  ? 

9.  What  per  cent,  profit  does  a  grocer  make  who  buys 
sugar  at  10  cents  and  sells  it  at  12  cents  ? 

10.  What  per  cent,  is  gained  on  an  article  bought  at  $3 
and  sold  at  $5  ? 

52S.  1.  A  dealer  sold  flour  at  a  profit  of  $2  a  barrel, 
and  gained  25%.     What  was  the  cost  ? 

Analysis.— Since  the  gain  was  25%  =  ^,  or  £,  $2  is  \  of  the 
cost ;  $2  is  £  of  4  times  $2,  or  $8.     Hence,  etc.    (516.) 

2.  Sold  hats  for  $1  less  than  cost,  and  lost  16-f %.  What 
did  they  cost  ? 

3.  A  merchant  sells  silk  at  a  profit  of  $1 J  a  yard,  which 
is  40%  gain.  What  did  it  cost,  and  what  is  the  selling 
price  ? 

4.  If  corn  selling  for  21  cents  a  bushel  more  than  cost 
gives  a  profit  of  30%,  what  did  it  cost  ? 

5.  Sold  sheep  at  $2£  more  than  cost,  which  was  a  profit 
of  50%.    What  did  they  cost,  and  what  is  the  selling  price  ? 

6.  Shoes  sold  at  $.50  above  cost  give  a  profit  of  12£%. 
What  did  they  cost  ? 

7.  A  farmer,  by  selling  a  cow  for  $12  less  than  she 
cost,  lost  33-J-%.     What  did  she  cost  ? 

8.  A  grocer  sells  a  certain  kind  of  tea  for  6  cents  a 
pound  more  than  cost  and  gains  5%.    What  did  it  cost  ? 


PROFIT     AND     LOSS.  279 

529.  1.  A  watch  was  sold  for  $120,  at  a  gain  of  20%. 
What  was  the  cost  ? 

Analysis.— Since  the  gain  was  20%,  or  £,  of  the  cost,  $120,  the 
selling  price,  is  f  of  the  cost.  |  of  $120,  or  $20,  is  |  of  the  cost,  and 
f,  or  the  cost  itself,  is  5  times  $20,  or  $100.    Hence,  etc.    (518.) 

2.  Sold  tea  at  ,$.  90  a  pound,  and  gained  25%.  What 
did  it  cost  ? 

3.  A  newsboy,  by  selling  his  papers  at  4  cents  each, 
gains  33£%.     What  do  they  cost  him  ? 

4.  A  man  sold  a  horse  and  harness  for  $330,  which  was 
10%  more  than  they  cost.     What  was  their  cost  ? 

5.  If  20%  is  lost  by  selling  wheat  at  $1.G0  a  bushel, 
what  would  be  gained  if  sold  at  20%  above  cost  ? 

6.  John  Rice  lost  40%  on  a  reaper,  by  selling  it  for  $60. 
For  what  should  he  have  sold  it  to  gain  40%  ? 

7.  If,  by  selling  books  at  $2  a  volume,  there  is  a  gain 
of  25%,  at  what  price  must  they  be  sold  to  lose  15%  ? 

8.  Two  pictures  were  sold  for  $99  each  ;  on  one  there 
was  a  gain  of  10%,  on  the  other  a  loss  of  10%.  Was  there 
a  gain  or  loss  on  the  sale  of  both,  and  how  much  ? 

WRITTEN     EXERCISES. 

530.  1.  A  hogshead  of  sugar  bought  for  $108.80  was 
sold  at  a  profit  of  12£%.     What  was  the  gain  ? 

Operation.— $108.80  x  .12|  =  $13.60.    (512.) 
Formula. — Profit  or  Loss  =  Cost  x  Rate  %. 

Find  the  Profit  or  Loss, 

2.  On  land  that  cost  $1745,  and  was  sold  at  a  gain  of  20%. 

3.  On  goods  that  cost  $3120,  and  were  sold  at  27%  gain. 

4.  On  a  boat  bought  for  $2545£,  and  sold  at  25%  loss. 


280  PERCENTAGE. 

5.  On  goods  bought  for  $2560.75,  and  sold  at  8%  loss. 

6.  On  25  tons  of  iron  rails  bought  at  $58  a  ton,  and 
sold  at  an  advance  of  17£%. 

7.  A  merchant  pays  $6840  for  a  stock  of  spring  goods, 
and  sells  them  at  an  advance  of  26-|-%  on  the  purchase 
price.    After  deducting  $3  75  for  expenses,  what  is  his  gain  ? 

8.  A  miller  bought  1000  bushels  of  wheat  at  $1.84  a 
bushel,  and  sold  the  flour  at  16f%  advance  on  the  cost  of 
the  wheat.     What  was  his  profit  ? 

9.  Bought  128  tons  of  coal  at  $5.12|  a  ton,  and  sold  it 
at  a  gain  of  22%.    What  was  the  entire  profit  ? 

10.  A  ship,  loaded  with  3840  bbl.  of  flour,  being  over- 
taken by  a  storm,  found  it  necessary  to  throw  37 i%  of  her 
cargo  overboard.     What  was  the  loss  at  $7.65  a  bbl.  ? 

11.  A  man  bought  a  pair  of  horses  for  $450,  which  was 
25%  less  than  their  real  value,  and  sold  them  for  25%  more 
than  their  real  value  ;  what  was  his  gain  ? 

531.  1.  Bought  a  house  for  $4380.    For  what  must  it 
be  sold  to  gain  14£%  ? 
Operation.— $4380  x  (1  +  .14|)  or  1.145  =  $5015.10.    (512.) 

2.  At  what  price  must  pork,  bought  at  $18.40  a  barrel, 
be  sold,  to  lose  15%? 

Operation.— $18.40  x  (1  -  .15),  or  .85  =  $15.64.    (512.) 

FoEMULA-SeUing  Price=  \  °0S*  X  £  +^te  *  of  Gain)' 

(  Cost  x  (1— Eate  %  of  Loss). 

Find  the  Selling  Price, 

3.  Of  goods  bought  at  $187.50,  and  sold  at  11%  gain. 

4.  Of  beef  bought  at  $12£  a  barrel,  and  sold  at  9£%  loss. 

5.  Of  cotton  bought  at  $.14,  and  sold  at  a  gain  of  21|%. 

6.  Of  cloth  that  cost  $5J  a  yard,  and  was  sold  at  a 
profit  of  18£%  ? 


PROFIT     AND     LOSS.  281 


7.  At  what  price  must  goods  that  cost  $3£  a  yard  be 


marked,  to  gain  25%  ?    To  lose 

8.  Sold  a  lot  of  damaged  goods  at  a  loss  of  15%.  What 
was  the  selling  price  of  those  that  cost  $.62-J-  ?    $1.25  ? 

9.  Bought  a  hogshead  of  sugar  containing  9  cwt.  56  lb. 
for  $86.04,  and  paid  $4.78  freight  and  cartage.  At  what 
price  per  pound  must  it  be  sold  to  gain  20%  ? 

532.  1.  Bought  wool  at  $.48  a  pound,  and  sold  it  at 
$.  60  a  pound.     What  per  cent,  was  gained  ? 

Operation.— $.60  -  $.48  =  $.12 ;  and  $.12  -f-  $.48  =  .25  =  25^. 
(515.) 

2.  Sold  for  $10.02  an  article  that  cost  $12.  What  was 
the  loss  per  cent.  ? 

Operation.— $12 -.$10.02 =$1. 98;  and$l  98-s-$12=.16£=16£%. 
Formula.— Rate  %  =  Profit  or  Loss  -t-  Cost. 
Find  the  rate  per  cent,  of  profit  or  loss, 

3.  On  sugar  bought  at  8  cents  and  sold  at  9J  cents. 

4.  On  tea  bought  at  $1,  and  sold  at  $.87£. 

5.  On  goods  that  cost  $275,  and  were  sold  for  $330. 

6.  On  grain  bought  for  $1.25  a  bushel,  and  sold  for 
$1.60  a  bushel. 

7.  On  a  sewing-machine  sold  for  $72.96,  at  again  of 
$9.12. 

8.  On  goods  sold  for  $425.98,  at  a  loss  of  $134.52. 

9.  Bought  paper  at  $3  a  ream,  and  sold  it  at  25  cents 
a  quire.     What  was  the  gain  per  cent.  ? 

10.  A  dealer  bought  108  bbl.  of  apples  at  $4. 62£,  and 
sold  them  so  as  to  gain  $114. 88 \.     What  was  his  gain  %? 

11.  If  \  of  an  acre  of  land  is  sold  for  £  the  cost  of  an 
acre,  what  is  the  gain  per  cent.  ? 


282  PERCENTAGE. 

12.  If  |  of  an  acre  of  land  is  sold  for  the  cost  of  \  of 
an  acre,  what  is  the  loss  per  cent.  ? 

13.  If  |  of  a  chest  of  tea  is  sold  for  what  the  whole 
chest  cost,  what  is  the  gain  per  cent,  on  the  part  sold  ? 

533.  1.  A  speculator  sold  grain  at  a  profit  of  33£$,  by 
which  he  made  25  cents  on  a  bushel.    What  did  it  cost  ? 

Operation.— $.25-^.33i=$.75.    Or,  $.25-*-i=$.75.    (518.) 

2.  Lost  $45. 75  on  the  sale  of  a  horse,  which  was  20$ 
of  the  cost.    What  was  the  cost  ? 

Operation.— $45.75-j- .20=$228. 75.    Or  $45.75-i-i=$228.75. 

Formula. — Cost  —  Profit  or  Loss  +  Rate  $. 

Find  the  Cost, 

3.  Of  goods  sold  at  $1500  profit,  or- a  gain  of  16$. 

4.  Of  flour  sold  at  a  loss  of  $.88,  or  10$,  on  a  barrel. 

5.  Of  wheat  sold  at  a  loss  of  6  cents,  or  4$,  on  a  bu.  ? 

6.  Of  lumber  sold  at  an  advance  of  $4.95  per  M.,  or 
35$  gain. 

7.  If  a  grocer  sells  his  stock  at  a  profit  of  15$,  what 
amount  must  he  sell  to  clear  $2500  ? 

8.  A  and  B  engage  in  speculation.  A  gains  $2000, 
which  is  12£$  of  his  capital,  and  B  loses  $500,  which  is 
5$  of  his  capital.     What  sum  did  each  invest? 

534.  1.  A  furniture  dealer  sold  two  parlor  sets  for 

$450  each  ;  on  one  he  made  15$,  on  the  other  he  lost  15$. 

What  did  each  cost  him  ? 

n  ( $450-s-(l  +  .15)=$391.30  + ,  cost  of  one. 

Operation.-  -j  $450^_(1_  <15)=|520.41  + ,  cost  of  the  other.  (520.) 

r,   *     on-       t>  ■     .    \  {^^r Rate %oi gain.) 
FoKUTJLA.-Cost=Selling  Prices  j  (1_Rate%of  te.) 


PROFIT     AND     LOSS.  283 

Find  the  Cost, 

2.  Of  coal  sold  at  $6,  being  at  a  loss  of  12J%. 

3.  Of  grain  sold  at  $.96  a  bushel,  at  a  gain  of  28%. 

4.  Of  silk  sold  for  $5.40  a  yard,  at  a  profit  of  10%. 

5.  Of  hops  sold  at  10  cents  a  pound,  at  a  loss  of  20$. 

6.  Of  fruit  sold  for  $207.48,  at  a  loss  of  15%. 

7.  Having  used  a  carriage  1  year,  I  sold  it  for  $125, 
which  was  25%  below  cost.  What  should  I  have  received 
had  I  sold  it  for  10%  above  cost  ? 

8.  B  sold  a  span  of  horses  to  C  and  gained  12|-% ;  C 
sold  them  to  D  for  $550,  and  lost  16$%.  What  did  the 
horses  cost  B  ? 

9.  If  a  piece  of  property  increases  in  value  each  year  at 
the  rate  of  25%  on  the  value  of  the  previous  year,  for  4 
years,  and  then  is  worth  $16000,  what  did  it  cost  ? 

535.  1.  Bought  cloth  at  $3.60  a  yard.  At  what  price 
must  it  be  marked  that  12£%  may  be  abated  from  the 
asking  price,  and  still  a  profit  made  of  16f  %  ? 

Operation  - i  Sellin9 Price    =$360 x &  +  -16!)=$4-20- 

'      t  J/ar^Pnc3=$420-Kl-.12i)=$180.    (519.) 

2.  At  what  price  must  shovels  that  cost  $1.12  each  be 
marked  in  order  to  abate  5%,  and  yet  make  25%  profit  ? 

3.  How  must  a  watch  be  marked,  that  cost  $120,  so 
that  4%  may  be  deducted  and  a  profit  of  20%  be  made  ? 

4.  A  merchant,  on  opening  a  case  of  goods  that  cost 
$.80  a  yard,  finds  them  slightly  damaged.  How  must  he 
mark  them,  to  fall  25%  in  his  asking  price,  and  sell  at  cost? 

5.  Bought  land  at  $60  an  acre  ;  how  much  must  I  ask 
an  acre,  that  I  may  deduct  25%  from  my  asking  price,  and 
still  make  20%  on  the  purchase  price  ? 


284  PERCENTAGE. 

COMMISSION. 

536.  An  Agent  or   Commission   Merchant 

is  a  person  who  buys  or  sells  merchandise,  or  transacts 
other  business  for  another,  called  the  Principal 

537.  Commission  is  the ,  fee,  or  compensation, 
allowed  an  agent  or  commission  merchant  for  transacting 
business,  and  is  usually  computed  at  a  certain  rate  per 
cent,  of  the  money  involved  in  the  transaction. 

538.  A  Consignment  is  a  quantity  of  goods  sent 
to  a  commission  merchant  to  be  sold. 

539.  The  Consignor  is  the  person  who  sends  the 
goods  for  sale.    A  consignor  is  sometimes  called  a  Shipper. 

540.  The  Consignee  is  the  person  to  whom  the 
goods  are  sent.     He  is  sometimes  called  a  Correspondent. 

541.  The  Net  Proceeds  of  a  sale  or  other  transac- 
tion is  the  sum  of  money  that  remains  after  all  expenses 
of  commission,  etc.,  are  paid. 

542.  A  Guaranty  is  security  given  by  a  commis- 
sion merchant  to  his  principal  for  the  payment  of  gooda 
sold  by  him  on  credit. 

543.  An  Account  Sales  is  a  written  statement 
made  by  a  commission  merchant  to  his  principal,  contain- 
ing an  account  of  goods  sold,  their  price,  the  expenses, 
and  the  net  proceeds. 

544.  A  Broker  is  a  person  who  buys  or  sell  stocks, 
bills  of  exchange,  real  estate,  etc.,  for  a  commission, 
which  is  called  Brokerage. 


COMMISSION.  285 

545.  The  principles  and  operations  of  Percentage  in- 
volved in  Commission  and  Brokerage  are  the  same  as 
those  already  treated. 

546.  The  following  are  the  corresponding  terms  : 

1.  The  Base  is  the  amount  of  sales,  money  invested, 
or  collected. 

2.  The  Bate  is  the  per  cent,  allowed  for  services. 

3.  The  Percentage  is  the  Commission  or  Broker- 
age. 

4.  The  Amount  or  Difference  is  the  amount  o" 
sales,  plus  or  minus  the  commission. 

WRITTEN    EXERCISES. 

547.  Find  the  Commission  or  Brokerage, 

1.  On  a  sale  of  flour  for  $2575,  at  %\%. 

Operation.— $2575  x  .035  =  $64.37£.   "(512.) 

Fokmula. — Amount  of  Sales  x  Rate  %  =  Commission. 

2.  On  the  purchase  of  a  farm  for  $13750,  at  2f$. 

3.  On  the  sale  of  a  mill  for  $9384,  at  \%. 

4.  On  the  sale  of  $21680  worth  of  wool,  at  \\%. 

5.  On  the  sale  of  250  bales  of  cotton,  averaging  520  lb., 
at  14f  cents  a  pound  ;  commission  \\%. 

6.  On  the  sale  of  175  shares  of  stock,  at  $92|  a  share ; 
brokerage,  \%. 

7.  On  the  sale  at  auction  of  a  house  and  the  furniture 
for  $9346.80,  at  6J#. 

8.  A  commission  merchant  sells  225  bbl.  of  potatoes 
at  $3.25  per  bbl.,  and  316  bbl.  of  apples  at  $4£  per  bbl. 
What  is  his  commission  at  4£$  ? 


286  PERCENTAGE. 

548.  Find  the  rate  of  commission  or  brokerage, 

1.  When  $89  commission  is  paid  for  selling  goods  for 
$3560. 

Operation.— 89  -z-  3560  =  .02|  =  2} % .    (515.) 

Formula. — Commission  -f-  Amount  of  Sales  ==  Bate  %. 

2.  When  $165  com.  is  paid  for  selling  goods  for  $4950. 

3.  When  $63  is  paid  for  collecting  a  debt  of  $1260. 

4.  When  $117.75  is  paid  for  selling  a  house  for  $7850. 

5.  When  $235.40  is  paid  for  buying  26750  lb.  of  wool 
at  32  cents  a  pound. 

6.  When  $125  is  paid  for  the  guaranty  and  sale  of  goods 
for  $2500. 

7.  Paid  my  N.  O.  agent  $74.25  for  buying  26400  lb.  of 
rice,  at  4£  ct.  a  lb.    What  was  the  rate  of  his  commission  ? 

549.  Find  the  Amount  of  Sales, 

1.  When  a  commission  of  $147  is  charged  at  o\%. 
Operation.— $147  -*-  .035  =  $4200.    (517.) 

Formula. — Commission  -r-  Rate  %  =  AmOunt  of  Sales. 

2.  When  $92. 80  commission  is  paid  at  3\%. 

3.  When  $210  commission  is  charged  at  6$. 

4.  When  $24  brokerage  is  paid  at  \%. 

5.  When  $135  commission  is  charged  at  \\%. 

6.  Paid  an  attorney  $72.03  for  collecting  a  note,  which 
was  a  commission  of  *l\%.    What  was  the  face  of  the  note  ? 

550.  Find  the  Amount  of  Sales, 

1.  When  the  net  proceeds  are  $4875,  commission  %\%. 
Operation.— $4875  -r-  .975  =  $5000.    (519.) 

Formula. — Net  proceeds  ~(1  —  Rate  %)=Amt.  of  Sales. 

2.  When  the  net  proceeds  are  $3281.25,  commission  12J$. 


commission-.  287 

3.  When  the  net  proceeds  are  $560,  and  the  com.  4$. 

4.  After  deducting  6%%  commission  and  $132  for 
storage,  my  correspondent  sends  me  $23654.25  as  the  net 
proceeds  of  a  consignment  of  pork  and  flour.  What  was 
the  gross  amount  of  the  sale  ? 

551.  Find  the  amount  to  be  invested, 

1.  If  $9500  is  remitted  to  a  correspondent  to  be  invest- 
ed in  woolen  goods,  after  deducting  5%  commission. 

Operation.— $9500  -r- 1.05  =  $9047.62.        (519.) 

Formula.— Amount  Remitted  -r-  (1  -f  Bate  %)  =  Sum 
Invested, 

2.  If  $4908  be  remitted,  deducting  ±\%  commission. 

3.  If  $3246.20  be  remitted,  .deducting  2%  commission. 

4.  If  $1511.25  be  remitted,  deducting  \%  commission. 

5.  If  $10701.24  be  remitted,  deducting  \%  brokerage. 

6.  A  dealer  sends  his  agent  in  Havana  $6720.80,  with 
which  to  purchase  oranges  and  other  fruits,  after  deduct- 
ing his  commission  of  5%.  What  sum  did  the  agent  invest, 
and  what  was  the  amount  of  his  commission  ? 

7.  What  amount  of  sugar  can  be  bought  at  8  cents  a 
pound,  for  $2523.40,  after  deducting  a  commission  of  A\%. 

8.  Remitted  to  a  stockbroker  $10650,  to  be  invested  in 
stocks,  after  deducting  \%  brokerage.  What  amount  of 
stock  did  he  purchase  ? 

9.  A  broker  received  $45337.50  to  invest  in  bond  and 
mortgage,  after  deducting  a  commission  of  %\%.  What 
amount  did  he  invest,  and  what  was  his  commission  ? 

10.  Sent  $250.92  to  my  agent  in  Boston,  to  be  invested 
m  prints  at  15  cents  a  yard,  after  taking  out  his  commis- 
sion of  %%,     How  many  yards  ought  I  to  receive  ? 


288  PERCENTAGE 


% 


REVIEW. 

ORjLL    exercises. 

552.  1.  If  stoves  bought  at  $36  each  are  sold  at  a 
profit  of  8$$,  what  is  the  gain? 

2.  What  will  be  the  expense  of  collecting  a  tax  of  $1000, 
allowing  5$  ? 

3.  What  will  a  broker  receive  for  selling  1600  worth  of 
stock,  at  f  $  brokerage  ? 

4.  A  man  having  $250  spent  $80.  What  per  cent,  of 
his  money  had  he  left  ? 

5.  If  a  man  sells  a  building  lot  that  cost  $300,  at  an 
advance  of  166$$,  what  is  his  gain  ? 

6.  $  of  30$  is  what  per  cent,  of  72$  ?  Of  144$?  Of 
180$?     240$? 

7.  Bought  a  horse  for  20$  less  than  $200,  and  sold  him 
for  10$  more  than  $200.    What  per  cent,  was  gained  ? 

8.  How  many  bushels  of  wheat  at  $2  a  busnel  can  an 
agent  buy  for  $2040,  and  retain  2$  on  what  he  expends 
as  his  commission  ? 

9.  If  by  selling  land  at  $150  an  acre  I  lose  25$,  how 
must  I  sell  it  to  gain  40$  ? 

10.  A  boy  bought  bananas  for  $3  a  hundred,  and  sold 
them  for  5  cents  each.     What  per  cent,  did  he  gain  ? 

11.  Bought  cannel  coal  at  $19  a  ton,  which  was  5$  less 
than  the  market  price.    What  was  the  market  price  ? 

12.  Paid  an  agent  $150,  or  a  commission  of  1J$,  for 
selling  my  house.     For  what  sum  was  the  house  sold? 

13.  If  an  article  is  sold  so  as  to  gain  £  as  much  as  it 
cost,  what  per  cent,  is  gained  ? 


^        BE  VIEW.  289 

14.  A  merchant  tailor  sold  some  linen  coats  at  $1.80 
each,  which  was  33£%  below  the  marked  price.  What 
was  the  marked  price  ? 

15.  A  grocer  bought  40  gal.  of  maple  syrup  at  the  rate 
of  4  gal.  for  $6,  and  sold  it  at  the  rate  of  5  gal.  for  $8. 
What  was  his  whole  gain,  and  his  gain  per  cent.  ? 

16.  How  much  wheat  must  a  farmer  take  to  mill  that 
he  may  bring  away  the  flour  of  4|  bushels,  after  the  miller 
takes  his  toll  of  10$  ? 

WRITTEN     EXERCISES. 

553.  1.  After  taking  out  lh%  of  the  grain  in  a  bin, 
there  remained  40  bu.  3i  pk.  How  many  bushels  were 
there  at  first  ? 

2.  The  net  profits  of  a  farm  in  2  years  were  $3485,  and 
the  profits  the  second  year  were  5%  greater  than  the 
profits  the  first  year.     What  were  the  profits  each  year  ? 

3.  A  has  32%  more  money  than  B  ;  what  per  cent,  less 
is  B's  money  than  A's  ? 

4.  Bought  450  bushels  of  wheat  at  $1.25  per  bushel,  and 
sold  it  at  $1.40  per  bushel.  What  was  the  whole  gain, 
and  the  gain  per  cent.  ? 

5.  A  man  drew  out  of  the  bank  -f  of  his  money,  and  ex- 
pended 30%  of  50%  of  this  for  728  bu.  of  wheat,  at  $1.12| 
a  bushel.     What  sum  had  he  left  in  bank  ? 

6.  Sold  goods  to  the  amount  of  $47649,  at  a  profit  of 
16f%.     Eequired  the  cost  and  the  total  gain. 

7.  A  broker  received  $37.50  for  selling  some  uncurrent 
money,  charging  \%  brokerage.     How  much  did  he  sell  ? 

8.  If  $  of  a  farm  is  sold  for  what  \  of  it  cost,  what  is 
the  gain  per  cent.  ? 

13 


290  PERCENTAGE 

9.  An  architect  charged  \%  for  plans  aifcf  specifications, 
and  l-f$  for  superintending  a  building  that  cost  $25000. 
What  was  the  amount  of  his  fee  ? 

10.  If  a  stationer  marks  his  goods  50$  above  cost,  and 
then  deducts  50$,  what  per  cent,  does  he  make  or  lose  ? 

11.  Sold  a  farm  for  $14700,  and  lost  12$.  What  per 
3ent.  should  I  have  gained  by  selling  it  for  $21000  ? 

12.  If  an  article  bought  at  20$  below  the  asking  price 
is  sold  at  16$  below  that  price,  what  is  the  rate  of  gain  ? 

13.  A  commission  merchant  sold  a  consignment  of 
goods  for  $5250,  and  charged  3£$  commission,  and  2£$ 
for  a  guaranty.     Find  the  net  proceeds. 

14.  Smith  &  Jones  bought  a  stock  of  groceries  for 
$13680.  They  sold  \  of  the  entire  stock  at  15$  profit,  \ 
at  18  f$,  \  at  20$,  and  the  remainder  at  33£$  profit.  What 
was  the  whole  gain,  and  the  average  gain  per  cent.  ? 

15.  Give  the  marking  prices  at  25$  advance,  of  the 
following  bill  of  goods,  and  the  amount  when  sold  at  a 
reduction  of  10$  from  those  prices : 

•      1  Case  of  Prints,  450  yd.,  @  $.12 

3  Pieces  Cassimeres,  65    "    @  3.25 

1  Bale  Ticking,  244   "    @    .20 

25  Dress  Shawls,  @  7.36 

1  Gr.  gross  Clark's  Thread,  144  doz.,  %    .70 

50  Gross  Buttons,  @  1.00 

16.  How  much  would  the  above  bill  of  goods  amount 
to  if  sold  at  5|$  below  a  marking  price  of  15$  above  cost  ? 

17.  What  would  be  the  net  proceeds  of  a  sale  of  18  cwt. 
75  lb.  of  sugar,  at  $9f  per  cwt.,  allowing  2£$  commission, 
and  $16  J  for  other  charges? 


COMMISSION.  291 

m 

18.  A  broker  jjHttves  $7125  to  invest  in  cotton,  at  Yi\ 
cents  a  pound.  If  his  commission  is  2|%  how  many 
pounds  of  cotton  can  he  buy  ? 

19.  If  the  sale  of  potatoes  at  $.75  a  barrel  above  cost 
gives  a  profit  of  18J $,  how  much  must  be  added  to  this 
price  to  realize  a  profit  of  31£#  ? 

20.  An  agent  in  Chicago  purchases  1000  bbl.  of  flour 
at  $6.80,  and  pays  5  cents  a  barrel  storage  for  30  days ; 
also,  3000  bu.  of  wheat  at  $1.20.  He  charges  a  commis- 
sion of  \\%  on  the  flour,  and  1  cent  a  bushel  on  the  wheat. 
What  sum  of  money  will  balance  the  account,  and  what  is 
the  amount  of  his  commission  ? 

21.  An  agent  in  Boston  received  28000  lb.  of  Texas 
cotton,  which  he  sold  at  $.12^  a  pound.  He  paid  $45.86 
freight  and  cartage,  and  after  retaining  his  commission, 
he  remits  his  principal  $3252.89  as  the  net  proceeds  of  the 
sale.     What  was  the  rate  of  his  commission  ? 

22.  The  following  bill  of  goods  was  sold  at  auction  : 
\\  bbl.  A  Sugar,  312  lb.,  %  $.12£  that  cost  %.\\\ 

\   «    Pulv.  "  96  " 

1  Chest  Y.  H.  Tea,       84  " 

1  Box  Soap,  60  " 

1  \  Sacks  Java  Coffee,  110  " 

184  lb.  Codfish, 

Allowing  a  commission  of  4^%  for  selling,  find  the  entire 

profit  or  loss,  and  the  gain  or  loss  per  cent,  on  the  whole. 

23.  A  merchant  in  New  York  imported  2400  yd.  of 
English  cloth,  for  which  he  paid  in  London  10s.  sterling 
a  yard,  and  the  total  expenses  were  $255.  He  sold  the 
cloth  for  $3.81  a  yard,  U.  S.  money.  What  was  his  whole 
gain,  and  his  gain  per  cent.  ? 


m 

ti 

"      .14 

1.10 

a 

"    1.12| 

.13 

a 

"      .10} 

.22^ 

a 

"      .24i 

.07^ 

a 

"      .081 

292 


PERCENTAGE. 


554. 


SYNOPSIS  FOR  REVIEW. 


1.  Definitions. 


2.  Elements. 


1.  Percentage.  2.  Per  Cent.  3.  Sign 
of  Per  Cent.  4.  Rate,  or  Rate  f0. 
5.  Base.  6.  Percentage.  7.  Amount. 
8.  Difference. 


j  1.  How  many  considered. 
it. 


How  many  must  be  given. 

3.  510.  1.  Principle.  2.  Rule.  3.  Formula. 

4.  513.  1.  Principle.  2.  Rule.  3.  Formula. 

5.  516.  1.  Principle.  2.  Rule.  3.  Formula. 

6.  519.  1.  Principle.  2.  Rule.  3.  Formula. 


Applications    op 
Percentage. 


8.  Profit  and  Loss.    4 


1.  Diff't  kinds.  ->        Without  Time. 


it 


9.  Commission, 


With  Time. 
2.  General  Rule. 

1.  Definition. 

2.  To  estimate  gains  and  losses. 

f  1.  Base. 
3    Correspond- J  2.  Bate. 
ing  terms.      j  3.  Percentage. 

^  4.  Am't  and  Biff. 

1.  Agent,  or  Com- 
mission Merchant 

2.  Commission. 

3.  Consignment, 
j  4.  Consignor. 

5.  Consignee. 

6.  Net  Proceeds. 

7.  Guaranty. 

8.  Account  Sales. 
,  9.  Broker. 

2.  Prin.  and  Operations  Involved. 

1.  Base. 

3.  Correspond-  J  2.  Bate. 

ing  terms.      1  3.  Percentage. 

k  4.  ^im'J  a»d  Ztojf. 


ORAL    EXERCISES  . 

555.  1.  When  5%  is  charged  for  the  use  of  money, 
how  many  dollars  should  be  paid  for  the  use  of  $100  Y 
For  the  use  of  $200  ?     Of  $500  ?     Of  $50  ? 

2.  At  K%  a  year,  what  should  be  paid  for  the  use  of 
$100  for  2  years  ?     Of  $200  for  3  years  ? 

3.  If  $500  is  loaned  for  3  years,  what  should  be  paid 
for  its  use,  at  b%  a  year  ?    At  6%  a  year  ? 

4.  If  I  borrow  $250,  and  agree  to  pay  ±%  a  year  for  its 
use,  how  much  will  be  due  the  lender  in  5  years  ? 

5.  If  $7  is  paid  for  the  use  of  $100  for  1  year,  what  is 
the  per  cent.  ? 

6.  If  $50  is  paid  for  the  use  of  $100  for  5  years,  what 
is  the  per  cent.  ? 

7.  If  $14  is  paid  for  the  use  of  $200  for  1  year,  what  is 
the  per  cent.  ? 

8.  At  6%,  what  decimal  part  of  the  money  borrowed  is 
equal  to  the  money  paid  for  its  use  ?   At  7%  ?   8%  ?    9f0  ? 

DEFINITIONS. 

556.  Interest  is  a  sum  paid  for  the  use  of  money. 

557.  The  Principal  is  the  sum  for  the  use  of 
which  interest  is  paid. 

558.  The  Mate  of  Interest  is  the  per  cent.,  or 
number  of  hundredths,  of  the  principal,  paid  for  its  use 
for  one  year. 


294 


PERCENTAGE, 


559.  The  Amount  is  the  sum  of  the  principal  and 
the  interest. 

560.  Legal  Interest  is  the  interest  according  to 
the  rate  per  cent,  fixed  by  law. 

561.  Usury  is  a  higher  rate  of  interest  than  is  al- 
lowed by  law. 

562.  The  legal  rates  of  interest  in  the  different  States 
are  as  follows : 


Name  of  State. 


Alabama 

Arkansas-1* 

Arizona 

California* 

Canada  and  Ireland 
Connecticut  .  .   . 

Colorado* 

Dakota 

Delaware 

Dist.  Columbia.  . 
England  and  France 

Florida* 

Georgia 

Idaho 

Illinois ... 

Indiana 

Iowa 

Kansas 

Kentucky 

Louisiana 

Maine* 

Maryland 

Massachusetts*.  . 
Michigan 


Rate. 


8% 
6% 

10% 

10% 
6% 
7% 

10% 
7% 
6% 
6% 
5% 
8% 
7% 

10% 
8% 
6% 
6% 
7% 
6% 
5% 
6% 
6% 
6% 
7% 


Any. 
Any. 

Any. 


Any. 

Any. 


10% 


Any. 
10% 


10% 
10% 
12% 
10% 
8% 
Any. 


Any. 

10% 


Name  of  State. 


Minnesota 

Mississippi 

Missouri 

Montana 

New  Hampshire. 

New  Jersey 

New  York 

North  Carolina. . 

Nebraska 

Nevada*.  .    ... 

Ohio 

Oregon 

Pennsylvania . . . 
Rhode  Island*.  . 
South  Carolina*. 

Tennessee 

Texas 

Utah* 

Vermont 

Virginia 

West  Virginia.  . 
Washington  T  * 

Wisconsin 

Wyoming 


Rate. 


7% 

12% 

6% 

10% 

6% 

10% 

10% 

6% 

H 

H 

6% 

8% 

10% 

15% 

10% 

Any. 

6% 

8% 

10% 

12% 

6% 



6% 

Any. 

7% 

Any. 

o% 

10% 

8% 

12% 

10% 

Any. 

6<& 

6% 

12% 

0% 

10% 

Any. 

7% 

10% 

12% 

1 .  When  the  rate  per  cent,  is  not  specified  in  accounts,  notes, 
mortgages,  contracts,  etc.,  the  legal  rate  is  always  understood. 

2.  Where  two  rates  are  specified,  any  rate  above  the  lower,  and 
not  exceeding  the  higher,  is  allowed,  if  stipulated  in  icriting. 

3.  In  the  States  marked  thus  (*)  the  rate  per  cent,  is  unlimited  if 
agreed  upon  by  the  parties  in  writing. 


INTEREST.  295 

563.  In  the  operations  of  interest  there  are  Jive  parts, 
or  elements,  namely  : 

The  Principal ;  the  Rate  per  Cent,  per  Annum  (for  one 
year)  :  the  Interest ;  the  Time  for  which  the  principal  is 
lent ;  and  the  Amount,  or  sum  of  the  Prin.  and  Int. 

564.  These  terms  correspond  respectively  to  Base, 
Rate,  Percentage,  and  Amount  in  Percentage,  excluding 
Time,  which  is  an  additional  element  in  Interest. 

Oil  AT,     EXERCISES. 

565.  1.  At  3%  for  1  yr.,  what  decimal  part  of  the  prin- 
cipal equals  the  interest  ?    At  b%  ?    At  8%  ?    At  12 \%  ? 

2.  What  is  the  interest  of  $20  for  1  year  at  5%  ? 
Analysis. — Since  the  interest  of  any  sum  at  5%  for  1  yr.  is  .05 

of  the  principal,  the  interest  of  $20  for  1  yr.  at  5  %  is  .05  of  $20,  or  $1. 

3.  What  is  the  interest  of  $50  for  1  yr.  at  5%?  G%?  1%? 

4.  What  is  the  interest  of  $80  for  1  yr.  at  K% ?  8%?  10%  ? 

5.  At  7%  for  5  yr.,  what  decimal  part  of  the  principal 

equals  the  interest  ? 

Analysis.— Since  the  interest  at  1%  for  1  yr.  is  .07  of  the  prin- 
cipal, the  interest  for  5  yr.  is  5  times  .07,  or  .35  of  the  principal. 
Or,  it  is  5  times  the  interest  for  1  year. 

6.  At  6%  for  3  yr.,  what  decimal  or  fractional  part  of 
the  principal  equals  the  interest  ?  At  7%  for  6  yr.  ?  At 
5%  for  5  yr.  ?     At  &±%  for  2  yr.  ?    At  10$  for  4  yr.  ? 

7.  Find  the  interest  of  $30  for  3  yr.  at  5%. 

Analysis.— Since  the  interest  of  any  sum  at  5%  for  1  yr.  is  .05 
of  the  principal,  for  3  yr.  it  is  .15,  and  .15  of  $30  is  $4.50.  Or,  the 
interest  for  1  yr.  is  .05  of  $30,  or  $1.50,  and  for  3  yr.  it  is  3  times  as 
much,  or  $4.50. 

8.  Find  the  int.  at  6%  of  $20  for  2  yr.    Of  $40  for  3  yr. 

9.  Find  the  int.  at  8%  of  $5  for  5  yr.     Of  $10  for  1 0  yr. 


296  PERCENTAGE. 

10.  At  8%  for  2  yr.  6  mo.,  what  decimal  part  of  the 
principal  equals  the  interest  ? 

Analysis. — Since  the  interest  of  any  sum  for  1  yr.  at  8%  is  .08  of 
the  principal,  the  interest  on  the  same  for  2  yr.  6  mo.  is  2^  times  .08, 
or  .20  of  the  principal.     Or,  it  is  2\  times  the  interest  for  1  year. 

11.  At  6%  for  3  yr.  3  mo.,  what  decimal  part  of  the 
principal  equals  the  interest  ?    At  9%  for  3  yr.  3  mo.  ? 

12.  Find  the  int.  of  $9  for  2  yr.  4  mo.  at  7%.     At  8%. 

13.  What  is  the  int.  of  $1000  for  2  yr.  3  mo.  at  10$  ? 
For  4  yr.  6  mo.  ?    For  5  yr.  3  mo.  ?    For  8  mo.  ? 

566.  Principle. — TJie  interest  is  the  product  of  three 
factors  ;  namely,  the  principal,  rate  per  annum,  and  time 
(expressed  in  years  or  parts  of  a  year). 

WRITTEN     EXERCISES. 

567.  To  find  the  interest  or  amount  of  any  sum, 
at  any  rate  per  cent.,  for  years  and  months. 

1.  Find  the  amount  of  $97.50,  at  1%,  for  2  yr.  6  mo. 

operation.  Analysis.— Since  the  interest  of 

$97.50  an7  sum  at  7%   for  1  yr.  is  .07  of 

Q7  the  principal,  the  interest  of  $97.50 

'- —  at  7%  for  1  yr.  is  .07  of  $97.50,  or 

$6.8250  Intforlyr  $6,825;  and  the  interest  for  2  yr, 

2£  6  mo.  is  2*  times  the  interest  for  1 

i>7  n«o-  T  .  ,    -      ,  yr.,  or  $17.06],  and  $17.06}- +  $97.50 

17.0620   Int.  for  2  yr.  6  mo.       *    L..*         A      A  * 

=  $114.564, tne  Amount. 
97.50        Principal. 


$114.5625   Amount. 

Find  the  interest  and  the  amount, 

2.  Of  $450  for  3  yr.  9  mo.  at-6%.    For  8  mo.  at  1%. 

3.  Of  $247  for  5  yr.  3  mo.  at  h\%.    For  10  mo.  at  8%. 

4.  Of  $500  for  4  yr.  2  mo.  at  10$.     For  llmo.  at  5%. 


INTEREST.  297 

Rule. — I.  Multiply  the  principal  by  the  rate,  and  the 
product  is  the  interest  for  1  year. 

II.  Multiply  the  interest  for  1  year  by  the  time  in  years, 
and  the  fraction  of  a  year ;  the  product  is  the  required 
interest. 

III.  Add  the  principal  to  the  interest  for  the  amount. 

Formula. — Interest  =  Principal  x  Rate  x  Time. 

Find  the  interest, 

5.  Of  $36.40  for  1  yr.  7  mo.  at  6%.    At  1%.    At  7£% 

6.  Of  $750.50  for  3  yr.  1  mo.  at  h%.     At  8%.     At  9%. 

7.  Of  $1346.84  for  2  yr.  4  mo.  at  %\%.    At  7i%. 

8.  Of  $138.75  for  4  yr.  3  mo.  at  10$.    At  12^. 

9.  Find  the  amount  of  $640  for  5  yr.  6  mo.  at  H%. 

10.  Find  the  amount  of  $56.64  at  8%  for  3  yr.  3  mo. 

11.  Made  a  loan  of  $1040  for  1  yr.  9  mo.  at  *l\%.  How 
much  is  due  at  the  end  of  the  time  ? 

12.  If  a  note  for  $375,  on  interest  at  8%,  dated  June  10, 
1874,  be  paid  Sept.  10,  1876,  what  amount  will  be  due? 

568.  To  find  the  interest  on  any  sum  of  money, 
for  any  time,  at  any  rate  per  cent. 

Obvious  Relations  between  Time  and  Interest. 
I.  The  interest  on  any  sum  for  1  year  at  1%  is  .01  of 
the  principal. 

It  is  therefore  equal  to  the  principal  with  the  decimal  point  re- 
moved two  places  to  the  left. 

II  The  interest  for  1  mo.  is  ^  of  the  interest  for  1  yr. 

III.  The  interest  for  3  days  is  -fa,  or  ^,  of  the  interest 
for  1  month  ;  hence  any  number  of  days  may  readily  be 
reduced  to  tenths  of  a  month  by  dividing  by  3. 


298  PERCENTAGE. 

IV.  The  interest  on  any  sum  for  1  month,  multiplied 
by  the  number  of  months  and  tenths  of  a  month  in  the 
given  time,  and  the  product  by  the  number  expressing 
the  rate,  will  be  the  required  interest. 


569. 1.  Find  the  int.  of  $361.20  for  1  yr.  3  mo.24  da.  at  7 

OPEKATION. 

$3,612       (01  of  the  Prin.)    Int.  for  1  yr.  at  1  %  (568,  I). 
.301       Int.  for  1  mo.  at  1%  (568,  II). 
15.8       Number  of  months  and  tenths  (568,  III). 


$4.7558       Int.  for  lyr.  3  mo.  24  da.  at  1$. 

7 


$33.2906  Int.  for  1  yr.  3  mo.  24  da.  at  1%  (568,  IV). 

What  is  the  interest, 

2.  Of  $137.25  for  1  yr.  6  mo.  10  da.  at  6%  ?    At  ±%  ? 

3.  Of  $510.50  for  3  yr.  7  mo.  15  da.  at  b%  ?    At  8%  ? 

4.  Of  $1297.60  for  2  yr.  11  mo.  18  da.  at  1%  ?    At  1$%? 

Rule. — I.  To  find  the  interest  for  1  yr.  at  \%. 
Remove  the  decimal  point  in  the  given  principal  two 
places  to  the  left 

II.  To  find  the  interest  for  1  mo.  at  1%. 
Divide  the  interest  for  1  year  by  12. 

III.  To  find  the  interest  for  any  time  at  1%. 
Multiply  the  interest  for  1  month  by  the  number  of 

months  and  tenths  of  a  month  in  the  given  time. 

IV.  To  find  the  interest  at  any  rate  %. 

Multiply  the  interest  at  \%for  the  given  time  by  the  num- 
ber expressing  the  given  rate. 

5.  Find  the  int.  of  $781.90  for  1  yr.  1  mo.  12  da.  at  1%. 

6.  Find  the  int.  of  $3000  for  11  mo.  21  da.  at  10$. 


INTEREST.  299 

7.  What  is  the  amt.  of  $1049  for  2  yr.  3  mo.  9  da.  at  §\%  ? 

8.  What  is  the  amt.  of  8216.75  for  3  yr.  5  mo.  11  da.  at  8%  ? 

9.  Required  the  int.  of  $250  from  Jan.  1,  1873,  to 
May  10,  1875,  at  7%. 

10.  Required  the  amount  of  $408.60  from  Aug.  20  to 
Dec.  18,  1876,  at  10$. 

11.  What  is  the  interest  on  a  note  for  $515.62,  dated 
March  1,  1873,  and  payable  July  16,  1875,  at  7%? 

12.  A  man  sold  his  house  and  lot  for  $12500 ;  the 
terms  were,  $4000  in  cash  on  delivery,  $3500  in  9  mo., 
$2600  in  1  yr.  6  mo.,  and  the  balance  in  2  yr.  4  mo.,  with 
6%  interest.     What  was  the  whole  amount  paid  ? 

570.  SIX  PER  CENT  METHOD. 
At  6%  per  annum,  the  interest  of  41 

For  12  mo is  6  cents,  or  .06  of  the  principal. 

"  2  "  or  -J-  of  12  mo.,  "1  cent,  ".01  " 
it  i  «  "^"12  "  "i  "  "  .005  " 
"  6  da."  £  "  1  "  "jV  "  "  .001  " 
"     1  "  "  £  "    6  da.    "  .00OJ-"  " 

571.  Principles. — 1.  The  interest  of  any  sum  at  6% 
is  one-half  as  many  hundredths  of  the  principal  as 
there  are  months  in  the  given  time. 

2.  The  interest  of  any  sum  at  6%  is  one-sixth  as 
many  thousandths  of  the  principal  as  there  are  days  in 
the  given  time. 

Thus,  the  interest  on  any  sum  at  6%  for  1  yr.  3  mo.,  or  15  mo., 
(s  £  of  .15,  or  .075,  of  the  principal ;  and  for  18  da.  it  is  \  of  .018, 
or  .003,  of  the  principal.  Hence,  for  1  yr.  3  mo.  18  da.,  it  is  .075 
+  .003  =  .078  of  the  principal. 

It  is  evident  that  an  odd  month  is  |  of  .01,  or  .005;  and  that 
any  number  of  days  less  than  6  is  such  a  fractional  part  of  .001  as 
the  days  are  of  6  days. 


300  PERCENTAGE. 

OMAL     EXERCISES. 

5K2.  What  is  the  interest, 

1.  Of  $1  at 6% for  1  year ?    2 years?    3  years?    5 years? 
8  years  ?     12  years  ? 

2.  Of  $1  at  ti%  for  1  month  ?   2  mo.  ?    3 'mo.  ?   4  mo.  ? 

5  mo.  ?    7  mo.  ?    9  mo.  ?    10  mo.  ?   15  mo.  ?  18  mo.  ? 
At  %%,  what  is  the  interest, 

3.  Of  $1  for  1  yr.  4  mo.  ?     1  yr.  7  mo.  ?     2  yr.  2  mo.  ? 

4.  Of  $1  for  1  day  ?    6  da.  ?    12  da.  ?    19  da.  ?    24  da.  ? 
33  da.  ?    36  da.  ?     45  da.  ?     63  da.  ? 

5.  Of  $1  for  1  mo.  12  da.  ?    For  3  mo.  15  da.  ?    For 

6  mo.  25  da.  ?     For  7  mo.  11  da.  ?    For  11  mo.  18  da.  ? 
Find  the  interest, 

6.  Of  11,  at  6%,  for  1  yr.  3  mo.  6  da.     For  1  yr.  9  mo. 
18  da.     For  1  yr.  5  mo.  19  da. 

7.  Of  $1  at  %%  for  2  yr.  1  mo.  9  da.    For  3  yr.  24  da. 

8.  Of  $1  at  6%  for  5  yr.  5  mo.  5  da.  For  4  yr.  7  mo.  10  da. 
At  6%,  find  the  interest, 

9.  Of  $1  for  2  yr.  6  mo.    Of  $2.     Of  $3.     Of  $5. 

10.  Of  II  for  4  yr.  2  mo.     Of  110.     Of  820.     Of  $30. 

11.  Of  15  for  1  yr.  4  mo.     For  2  yr.     For  2  yr.  8  mo. 

12.  Of  $1  for  33  da.    For  63  da.    For  93  da.    For  123  da. 

13.  Of  16  for  33  da.     Of  14  for  63  da.     Of  12  for  93  da. 

14.  If  the  interest  of  a  certain  principal  at  §%  is  118, 
what  would  the  interest  be  at  b% ?    7%?    S%?    9% ?. 

5%  is  £  less  than  6%  ;  7%  is  £  more  than  6%  ;  Sfo  is  \  more,  etc. 

15.  If  the  interest  of  a  certain  principal  is  116,  what 
would  the  int.  be  at  3%?    4$%?    5%?    1$%?    S%?    12%? 

16.  If  the  interest  of  a  certain  principal  is  130,  what 
would  the  int.  be  at  2^?    4$?    7^?    S%?    10#?    U%? 


INTEREST.  301 


WRITTEN      EXERCISES. 

573.  1.  What  is  the  int.  of  $427.20  at  6%  for  2  yr.  5  mo. 
27  da? 

pekation.  Analysis. — Since  the  in- 

2  yr.  5  mo.  =  29  mo.       $427.20  terest  of  $x  for  2  ?T-  5  mo- 

7  27  da.  is  $.149  \>  or  of  any 

J  of  .29        =  .145               .149^  sum  is  .1491  of  the  prima* 

i  of  .027     =  .004^       $63.8664  pal  (571),  $427.20  x  .149* 

Int.    =  049Tof  the  Prin.  =$63.866+ is  the  required 

*  interest. 


Find  the  interest  at  6%  of 


2.  $597.25  for  7  mo.  18  da. 

3.  $418.75  for  1  mo.  25  da. 

4.  $309.18  for  2  yr.  24  da. 


5.  $1298  for  3  yr.  1  mo.  13  da. 

6.  $2000  for  2  yr.  7  mo.  24  da. 

7.  $4010  for  1  yr.  1  mo.  13  da. 


Eule. — Multiply  the  given  principal  by  the  decimal  ex- 
pressing the  interest  of  $1 ;  or  by  the  decimal  expressing 
one-half  as  many  hundredths  as  there  are  months,  and  one- 
sixth  as  many  thousandths  as  there  are  days,  in  the  given 
time,  and  the  product  ivill  be  the  required  interest. 

To  find  the  interest  at  any  other  per  cent,  by  this  method,  increase 
or  diminish  the  interest  at  6%  by  such  part  of  itself  as  the  given 
rate  is  greater  or  less  than  6%. 

574.  To  compute  Accurate  Interest,  that  is, 
reckoning  365  da.  to  the  year,  use  the  following 

Rule. — Find  the  interest  for  years  and  aliquot  parts  of 
a  year  by  the  common  method,  and  for  days  take  such  part 
of  1  year's  interest  as  the  number  of  days  is  of  365.     Or, 

When  the  time  is  in  days  and  less  than  1  year,  find  the 
interest  by  the  common  method,  and  then  subtract  -fa  part 
of  itself  for  the  common  year,  or  -fc,  if  it  be  a  leap  year. 


302  PERCENTAGE. 

1.  Find  the  accurate  interest  of  $1560  for  45  da.  at  7%. 
The  exact  int.  of  $1560  for  45  da.  at  7  %  =  $109-20  x  4>  =  13.46+. 

OOO 

Or,  it  is  $13.65  -  $18^  *  1  =  $13.46  +. 

Jo 

2.  Find  the  exact  int.  of  $1600  for  1  yr.  3  mo.  at  6$. 

3.  What  is  the  difference  between  the  exact  interest  of 
$648.40  at  8%  for  1  yr.  3  mo.  20  da.  and  the  interest 
reckoned  by  the  6%  method? 

4.  Find  the  exact  interest  of  $875.60  at  7%  for  63  da. 

5.  Required  the  exact  interest  on  three  U.  S.  Bonds  of 
$1000  each,  at  6%,  from  May  1  to  Oct.  15. 

6.  What  is  the  exact  interest  on  a  $500  U.  S.  Bond,  at 
5%,  from  Nov.  1  to  April  10  following  ? 

515.  Find  the  interest,  by  any  of  the  ordinary  methods, 

1.  Of  $721.56  for  1  yr.  4  mo.  10  da.  at  6%. 

2.  Of  $54.75  for  3  yr.  24  da.  at  5%. 

3.  Of  $1000  for  11  mo.  18  da.  at  7%. 

4.  Of  $3046  for  7  mo.  26  da,  at  8%. 

5.  Of  $1821.50  from  April  1  to  Nov.  12  at  6$. 

6.  Of  $700  from  Jan.  15  to  Aug.  1  at  10$. 

7.  Of  $316.84  from  Oct.  20  to  March  10  at  7%. 

What  is  the  amount 

8.  Of  $3146  for  2  yr.  3  mo.  10  da.  at  7%  ? 

9.  Of  $96.85  for  3  yr.  1  mo.  27  da.  at  6%  ? 

10.  Of  $1008.80  for  10  mo.  16  da.  at  6£$? 

11.  Of  $2000  for  15  da.  at  12^? 

12.  Of  $137.60  for  127  da.  at  10$? 

13.  If  $1671.64  be  placed  at  interest  June  1,  1874,  what 
amount  will  be  due  April  1,  1876,  at  7% •? 


INTEREST.  303 

14.  How  much  is  the  interest  on  a  note  for  $600,  dated 
Feb.  1, 1872,  and  payable  Sept.  25,  1875,  at  6%  ? 

15.  If  a  man  borrow  $9700  in  New  York,  and  loan  it 
in  Colorado,  what  will  it  gain  at  legal  int.  in  a  year  ? 

16.  Required  the  interest  of  $127.36  from  Dec.  12, 1873, 
to  July  3,  1875,  at  ±\%. 

17.  A  note  of  $250,  dated  June  5,  1874,  was  paid  Feb, 
14,  1875,  with  interest  at  8%.     What  was  the  amount  ? 

18.  A  note  for  $710.50,  with  interest  after  3  mo.,  at  1%, 
was  given  Jan.  1,  1874,  and  paid  Aug.  12,  1876.  What 
was  the  amount  due  ? 

19.  A  man  engaged  in  business  was  making  12£%  an- 
nually on  his  capital  of  $16840.  He  quit  his  business 
and  loaned  his  money  at  1\%.  What  did  he  lose  in  2  yr. 
3  mo.  18  da.  by  the  change  ? 

20.  A  man  borrows  $2876.75,  which  belongs  to  a  minor 
who  is  16  yr.  5  mo.  10  da.  old,  and  he  is  to  retain  it  until 
the  owner  is  21  years  old.  What  will  then  be  due  at  8% 
simple  interest  ?  • 

21.  A  speculator  borrowed  $9675,  at  6%,  April  15, 1874, 
with  which  he  purchased  flour  at  $6.25.  May  10,  1875, 
he  sold  the  flour  at  $7|  a  barrel,  cash.  What  did  he  gain 
by  the  transaction  ? 

22.  A  man  borrows  $10000  in  Boston  at  6$,  reckoning 
360  da.  to  the  year,  and  lends  it  in  Ohio  at  8%,  reckoning 
365  da.  to  the  year.     What  will  be  his  gain  in  146  days  ? 

23.  A  tract  of  land  containing  450  acres  was  bought  at 
$36  an  acre,  the  money  paid  for  it  being  loaned  at  5£$. 
At  the  end  of  3  yr.  8  mo.  24  da.,  f  of  the  land  was  sold 
at  $40  an  acre,  and  the  remainder  at  $38-|  an  acre.  What 
was  gained  or  lost  by  the  transaction  ? 


304  PERCENTAGE. 

PROBLEMS  IN  INTEREST. 

576.  Interest,  time,  and  rate  given,  to  find  the 
principal. 

ORAL    EXERCISES. 

1.  What  sum  of  money  will  gain  $10  in  1  yr.  at  6%? 

Analysis.— The  interest  of  $1  for  1  yr.  at  5%  is  .05  of  the  prin- 
cipal, and  therefore  $10  -f-  .05,  or  $200,  is  the  required  sum.    Or, 

Since  $.05  is  the  interest  of  $1,  $10  is  the  interest  of  as  many 
dollars  as  $.05  is  contained  times  in  $10,  or  200  times.     Hence,  etc. 

What  sum  of  money  will  gain, 
2.  $20  int.  in  2  yr.  at  5%  ?     5.  $84  int.  in  2  yr.  at  7%  ? 


3.  $25  int.  in  5  yr.  at  5%  ? 

4.  $00  int.  in  2  yr.  at  6%? 


6.  $50  int.  in  6  mo.  at  10$  ? 

7.  $30  int.  in  3  mo.  at  S%? 


WRITTEN    EXERCISES. 

577.  1.  What  sum  of  money,  put  at  interest  3 \  yr.  at 
6%,  will  gain  $346.50? 

OPERATION. 

Int.  of  $1  for  S £  yr.  at  6%  =  $.21.        Analysis.— Same  as  in 
$346.50  +  $.21  =  1650  times  ;  oral  exercises-    (5™.) 

$1  x  1650  -  $1650. 

What  principal 

2.  Will  gain  $49.50  in  1  yr.  3  mo.  at  6%  ?    At  5%  ? 

3.  Will  gain  $153.75  in  3  mo.  24  da.  at  7%  ?    At  S%? 
Kule. — Divide  the  given  interest  by  the  interest  of  $1 

for  the  given  time,  at  the  given  rate. 
Formula.— Principal  =  Interest  -r-  (Bate  x  Time). 

What  sum  of  money 

4.  Will  gain  $213  in  5  yr.  10  mo.  20  da.  at  7%  ? 

5.  Will  gain  $1 73.97  in  4  yr.  4  mo.  at  6%?    At  12$? 


INTEREST.  305 

6.  A  man  receives  semi-annually  $350  int.  on  a  mort- 
gage at  1%.     What  is  the  amount  of  the  mortgage  ? 

578.  Amount,  rate,  and  time  given,  to  find  the 
principal. 

ORAL    EXERCISES. 

1.  What  sum  of  money  will  amount  to  $107  in  1  yr. 

at  7%? 

Analysis. — Since  the  interest  is  .07  of  the  principal,  the  amount 
is  1.07,  or  |fj,  of  it.  If  $107  is  {%%  of  the  principal,  T^  of  the  prin- 
cipal is  T^j  of  $107,  or  $1 ;  and  f £$,  or  the  principal  itself,  is  $100.  Or, 

Since  $1.07  is  the  amount  of  $1,  $107  is  the  amount  of  as  many 
dollars  as  $1.07  is  contained  times  in  $107,  or  $100. 

What  sum  of  money  will  amount  to 


2.  $130  in  5  yr.  at  6%? 

3.  $228  in  2  yr.  at  7$  ? 

4.  $412  in  6  mo.  at  6%  ? 


5.  $250  in  10  yr.  at  10$? 

6.  $350  in  15  yr.  at5$? 

7.  $260  in  3  yr.  9  mo.  at  8%  ? 


WRITTEN    EXERCISES. 

579.  1.  What  sum  will  amount  to  $3^.50  in  5  yr. 

OPERATION. 

Am't  of  $1  for  5  yr.  at  1%  =  $1.35.       Analysis.  —  Same  as 
$337.50  -v-  $1.35  =  250  times  J  in  oral  exercises.  (578.) 

$1  x  250  =  $250. 

What  principal 

2.  Will  amount  to  $1028  in  4  mo.  24  da.  at  7$? 

3.  Will  amount  to  $1596  in  2  yr.  6  mo.  at  h\%  ? 

4.  Will  amount  to  $1531.50  in  3  mo.  18  da.  at  7$? 

Rule. — Divide  the  given  amount  ly  the  amount  of  $1 
for  the  given  time,  at  the  given  rate. 

Formula. — Prin.  =  Amt-~  (1  4-  Rate  x  Time). 


306  PERCENTAGE. 

5.  What  is  the  principal  which  in  217  days,  at  5iJ% 
amounts  to  $918.73  ? 

6.  What  principal  in  3  yr.  4  mo.  24  da.  will  amount 
to  $761.44  at  5%? 

580.  Principal,  interest,  and  time  given,  to  find 
the  rate. 

ORAIj    E XE RCISES. 

1.  At  what  rate  will  $100  gain  $14  in  2  years  ? 

Analysis. — Since  the  interest  of  $100  is  $14  for  2  yr.,  it  is  $7  for 
1  yr.,  and  $7  is  .07  of  $100,  the  principal.   Hence  the  rate  is  7  % .   Or, 

Since  the  interest  of  $100  for  2  yr.  at  1  %  is  $2,  $14  is  as  many 
per  cent,  as  $2  is  contained  times  in  $14,  or  7$. 


At  what  rate  will 

2.  $300  gain  $60  in  4  yr.  ? 

3.  $500  gain  $100  in  5  yr.  ? 

4.  $400  gain  $84  in  3  yr.  ? 


5.  $5  gain  $1  in  3  yr.  ? 

6.  $120  gain  $60  in  10  yr.  ? 

7.  $150  double  itself  in  10  yr.? 


WRITTEN    EXERCISES. 

« 

581.  1.  At  what  rate  per  cent,  will  $1600  gain  $280 
interest  in  2 \  years  ? 

OPERATION. 

Int.  of  $1600  at  1%  for  24  yr.  =  $40.      ANALYsis.-Same  as 

in      oral       exercises. 
$280  -~  $40  =  7  times  ;  1%  x  7 = 1%.    (580#) 

At  what  rate  per  cent 

2,  Will  $2085  gain  $68.11  in  5  mo.  18  da.  ? 

3.  Will  $1500' gain  $252  in  2  yr.  4  mo.  24  da.  ? 

Rule. — Divide  the  given  interest  by  the  interest  of  the 
given  principal,  for  the  given  time,  at  1%. 

Formula. — Rate  =  Int.  -~  (Prin.  x\%  x  Time). 


INTEREST.  307 

4.  A  house  that  cost  $14500  rents  for  $1189.  What  per 
cent,  does  it  pay  on  the  investment : 

5.  At  what  rate  will  $1500  amount  to  $1684.50  in 
2  yr.  18  da,  ? 

6.  At  what  rate  per  month  will  $2000  gain  $120  in 
90  da.  ? 

7.  A  man  invests  $15600,  which  gives  him  an  annual 
income  of  $1620.     What  rate  of  interest  does  he  receive  ? 

8.  At  what  rate  per  annum  will  any  sum  double  itself 
in  4,  6,  8,  and  10  years,  respectively  ? 

ki\c/0,  any  sum  will  double  itself  in  100  yr.  ;  hence,  to  double 
itself  in  4  yr.,  the  rate  will  be  as  many  times  l/fl  as  4  yr.  are  con- 
tained times  in  100  yr ,  or  25#>,  etc. 

9.  At  what  rate  per  annum  will  any  sum  triple  itself 
in  2,  5,  7,  12,  and  20  years,  respectively  ? 

10.  I  invest  $49500  in  a  business  that  pays  me  $297  a 
month.     What  annual  rate  of  interest  do  I  receive  ? 

11.  Which  is  the  better  investment,  and  how  much, 
one  of  $4200,  yielding  $168  semi-annually,  or  one  of 
$7500,  producing  $712£  annually  ? 

582.  Principal,  interest,  and  rate  given,  to  find 
the  time. 

ORAL      EXERCISES. 

1.  In  what  time  will  $200  gain  $56  at  7$  ? 

"-» 

Analysis.— The  given  interest,  $56,  is  -$%,  Or  .28,  of  the  princi- 
pal,  $200;  therefore,  the  time  is  as  many  years  as  .07,  the  given 
rate,  is  contained  times  in  .28,  or  4  times.     Hence,  etc. 

Or,  the  interest  of  $200  at  7%  for  1  yr.  is  $14;  therefore,  the 
time  is  as  many  years  as  $14  are  contained  times  in  the  given  inter- 
est, $56,  or  4  years.     Hence,  etc. 


30.8  PERCENTAGE. 


In  what  time  will 

2.  $40  gain  $10  at  b%  ? 

3.  $500  gain  $100  at  4$  ? 

4.  $25  gain  $20  at  6%? 


5.  $1000  gain  $250  at  5%  ? 

6.  $5  gain  90  cents  at  6%  ? 

7.  $50gain$12iatl0^? 


WRITTEN    EXERCISES. 

583.  1.  In  what  time  will  $840  gain  $78.12  at  6%? 

OPERATION. 

$840  x  .06 = $50.40  Int.  for  1  yr.       Analysis.-  Same  as  in  the 
$78.12^$50.40=1.55.  oral  exercises.     (582.) 

1  yr.  x  1.55=1  yr.  6  mo.  18  da. 

In  what  time 

2.  Will  $175.12  gain  $6.43  at  6%? 

3.  Will  $1000  amount  to  $1500  at  7£%? 

Rule. — Divide  the  given  interest  by  the  interest  of  the 
given  principal,  at  the  given  rate  for  1  year. 
Formula. — Time  =  Interest  -f-  (Prin.  x  Rate). 

4.  In  what4ime  will  $8750  gain  $1260  at  %%  a  month? 

5.  How  long  must  $1301.64  be  on  interest  to  amount 
to  $1522.92  at  5%? 

6.  How  long  will  it  take  any  sum  of  money  to  double 
itself  at  Z%,  5%,  6%,  7|%  and  10%',  respectively? 

At  100%,  any  sum  of  money  will  double  itself  in  1  year;  hence 
to  double  itself  at  10%,  it  will  require  as  many  years  as  10  fo  is 
contained  times  in  100%,  or  10  yr. 

7.  How  long  will  it  take  any  sum  to  triple  itself  at 
t%>  5%  "!%  %%,  and  Vfy%,  respectively  ? 

8.  In  what  time  will  the  interest  of  $120,  at  8$,  equal 
the  principal  ?  Equal  half  the  principal  ?  Equal  Uvice 
the  principal  ? 


INTEREST.  309 


COMPOUND    ESTTEEEST. 

584c.  Compound  Interest  is  interest  not  only  on 
the  principal,  but  on  the  interest  added  to  the  principal 
when  it  becomes  due  ? 

ORAL      EXERCISES. 

585.  1.  What  is  the  comp.  int.  of  $500  in  2  yr.  at  6%  ? 

Analysis. — The  simple  interest  of  $500  for  2  yr.  is  $60  ;  the  in- 
terest of  the  first  year's  interest,  $30,  for  the  second  year  is  $1.80, 
which,  added  to  $60,  gives  $61.80,  the  compound  interest    Or, 

The  interest  of  $500  for  1  yr.  at  6%  is  $30,  and  the  amount  is 
$530,  which  is  the  principal  for  the  second  year;  the  interest  of  $530 
for  1  yr.  at  6%  is  $31.80,  which  added  to  $530  gives  $561.80,  the 
final  amount ;  and  deducting  $500,  the  original  principal,  gives 
$61.80,  the  compound  interest. 

What  is  the  compound  interest 

2.  Of  $600  for  2  yr.  at  5%  ?      4.  Of  $300  for  2  yr.  at  10$? 

3.  Of  $100  for  2  yr.  at  1%  ?     5.  Of  $1000  for  2  yr.  at  5%? 
What  is  the  amount  at  compound  interest, 


6.  Of  $800  for  2  yr.  at  5%  ? 

7.  Of  $2000  for  2  yr.  at  10$? 


8.  Of  $400  for  2  yr.  at  4%  ? 

9.  Of  $500  for  2  yr.  at  8%  ? 


WRITTEN     EXAMPLES.' 

586.  1.  What  is  the  comp.  int.  of  $750  for  2  yr.  at  6%? 

opekation.  Analysis.— Since  the  amount  is  1.06 

$750   Prin.  for  1st  yr.     of  the  principal,  the  amount  at  the  end 

1.06  of  the  first  year  is  $795,  which  is  the 

awo-    „  .     .  principal  for  the  2d  year,  and  the  amount 

$790    Prm.  for  Myr.      ^  ^  ^    q{  ^  £   ^  ^   ^^ 

1-^"  Hence,  by  subtracting  the  given  princi- 

$842.70   Total  amount       Pa^>  $750,  the  result  is  the  compound 
75Q  interest,  $92.70. 

$92.70    Compound  int. 


310  PERCENTAGE. 

2.  What  will  $350  amt.  to  in  3  yr.  at  1%  comp.  int.  ? 

3.  What  is  the  compound  int.  of  $1200  for  3  yr.  at  5%  ? 

Eule. — I.  Find  the  amount  of  the  given  principal  for 
the  first  period  of  time  at  the  end  of  which  interest  is  due, 
and  make  it  the  principal  for  the  second  period. 

II.  Find  the  amount  of  this  principal  for  the  next  period; 
and  so  continue  till  the  end  of  the  given  time. 

III.  Subtract  the  given  principal  from  the  last  amount, 
and  the  remainder  will  he  the  compound  interest. 

When  the  time  contains  months  and  'days,  less  than  a  single 
period,  find  the  amount  up  to  the  end  of  the  last  period,  and  com- 
pute the  simple  interest  upon  that  amount  for  the  remaining  months 
and  days,  which  add  to  find  the  total  amount. 

4.  What  will  $864.50  amount  to  in  4  yr.  at  8%,  com- 
pound interest  ? 

5.  What  is  the  compound  interest  of  $680  for  2  yr.  at 
11%,  interest  being  payable  semi-annually  ? 

6.  What  is  the  compound  interest  of  $460  for  1  yr. 
5  mo.  18  da.  at  6%,  interest  payable  quarterly  ? 

7.  What  will  be  the  amount  of  $1250  in  3  yr.  7  mo. 
18  da.  at  5%,  interest  being  semi-annual  ? 

8.  Find  the  compound  interest  of  $790  for  9  mo.  27  da. 
at  8%,  payable  quarterly. 

The  computation  of  compound  interest  may  be  abridged  by 
using  the  following  table 

To  use  the  table,  multiply  the  given  principal  by  the  number  in 
the  table  corresponding  to  the  given  number  of  years  and  the  given 
rate.  If  the  interest  is  not  annual,  reduce  the  time  to  periods,  and 
the  rate  proportionally.  Thus,  2  yr.  6  mo. ,  by  semi-annual  payments, 
at  7%,  is  the  same  as  5  yr.  at  %\%  ;  and  1  yr.  9  mo.,  quarterly 
payments,  at  8%,  the  same  as  7  yr.  2X2%. 


INTEREST. 


311 


581.  Table  showing  the  ami.  of  $1,  at  2 \,  3,  Z\,  4,  5,  6,  7, 
8,  9, 10,  11,  and  12%,  compound  int.,  from  1  to  20  years. 


Yrs.  2$  percent. 

3  per  cent. 

Si  per  cent. 

4  per  cent. 

5  per  cent. 

6  per  cent. 

1 

1.025000 

1.030000 

1.035000 

1.040000 

1.050000 

1.060000 

2 

1.050625 

1.060900 

1.071225 

1.081600 

1.102500 

1.123600 

3 

1.076891 

1.092727 

1.108718 

1.124864 

1.157625 

1.191016 

4 

1.103813 

1.125509 

1.147523 

1.169859 

1.215506 

1.262477 

5 

1.131408 

1.159274 

1.187686 

1.216653 

1.276282 

1.338226 

6 

1.159693 

1.194052 

1.229255 

1.265319 

1.340096 

1.418519 

7 

1.188686 

1.229874 

1.272279 

1.315932 

1.407100 

1.503630 

8 

1.218403 

1.266770 

1.316809 

1.368569 

1.477455 

1.593848 

9 

1.248863 

1.304773 

1.362897 

1.423312 

1.551328 

1.689479 

10 

1.280085 

1.343916 

1.410599 

1.480244 

1.628895 

1.790848 

11 

1.312087 

1.384234 

1.459970 

1.539454 

1.710339 

1.898299 

12 

1.344889 

1.425761 

1.511069 

1.601032 

1.795856 

2.012197 

13 

1.378511 

1.468534 

1.563956 

1.665074 

1 .885649 

2.132928 

14 

1.412974 

1.512590 

1.618695 

1.731676 

1.979982 

2.260904 

15 

1.448298 

1.557967 

1.675349 

1.800944 

2.078928 

2.396558 

16 

1.484506 

1.604706 

1.733986 

1.872981 

2.182875 

2,540352 

17 

1.521618 

1.652848 

1.794676 

1.947901 

2.292018 

2.692773 

18 

1  559659 

1.702433 

1.857489 

2.025817 

2.406619 

2.854339 

19 

1.598650 

1.753506 

1.922501 

2.106849 

2.526950 

3.025600 

20 

1.638616 

1.806111 

1.989789 

2.191123 

2.653298 

3.207136 

Yrs. 

7  per  cent. 

8  per  cent. 

9  per  cent. 

10  per  cent. 

11  per  cent. 

12  per  cent. 

1 

1 .070000 

1.080000 

1.090000 

1.100000 

1.110000 

1.120000 

2 

1.144900 

1.166400 

1.188100 

1.210000 

1.232100 

1.254400 

3 

1.225043 

1.259712 

1.295029 

1.331000 

1.367631 

1.404908 

4 

1  310796 

1.360489 

1.411582 

1.464100 

1.518070 

1.573519 

5 

1.402552 

1.469328 

1.538624 

1.610510 

1.685058 

1.762342 

6 

1.500730 

1.586874 

1  677100 

1.771561 

1.870414 

1.973822 

7 

1.605781 

1.713824 

1.828039 

1.948717 

2.076160 

2.210681 

8 

1.718186 

1.850930 

1.992563 

2.143589 

2.304537 

2,475963 

9 

1.838459 

1.999005 

2.171893 

2.357948 

2.558036 

2.773078 

10 

1.967151 

2.158925 

2.367364 

2.593742 

2.839420 

3.105848 

11 

2.104852 

2.331639 

2.580426 

2.853117 

3.151757 

3  478549 

12 

2.252192 

2.518170 

2.812665 

3.138428 

3.498450 

3  895975 

13 

2.409845 

2.719624 

3.065805 

3.452271 

3.883279 

4.363492 

14 

2.578534 

2.937194 

8.341727 

3.797498 

4.310440 

4.887111 

15 

2.759031 

3.172169 

3.642482 

4.177248 

4.784588 

5.473565 

16 

2.952164 

3.425943 

3.970306 

4.594973 

5.310893 

6.130392 

17 

3.158815 

3.700018 

4.327633 

5.054470 

5.895091 

6.866040 

18 

3.379932 

3.996019 

4.717120 

5.559917 

6.543551 

7.689964 

19 

3.616527 

4.315701 

5.141661 

6.115909 

7.263342 

8.612760 

20 

3.869684 

4.660957 

5.604411 

6.727500 

8.062309 

9.646291 

312  PERCENTAGE. 

9.  Find  by  the  table  the  compound  interest  of  $950  for 
lyr.  5  mo.  24  da.,  at  10%,  interest  payable  quarterly. 

OPERATION. 

1  yr.  5  mo.  24  da.  =  5  quarters  of  a  year +  2  mo.  24  da. 
10%  per  annum  =  2\%  per  quarter. 
Amount  for  5  yr.  at  2\%  =  1.131408  of  principal. 
$950  x  1.131408  =  $1074.887,  amount  for  1  yr.  3  mo. 
Interest  of  $1074.837  for  2  mo.  24  da.  at  10%  =  $25,079. 
$1074.837 +  $25,079  =  $1099.916,  total  amount. 
$1099.910  -  $950  =  $149,916,  compound  interest. 

10.  Find  the  amount,  at  compound  interest,  of  $749.25 
for  10  yr.  4  mo.,  at  7%,  interest  payable  semi-annually. 

11.  What  sum  placed  at  simple  interest  for  3  yr.  10  mo. 
18  da.,  at  7%,  will  amount  to  the  same  as  $1500  placed  at 
compound  interest  for  the  same  time,  and  at  the  same 
rate,  payable  semi-annually  ? 

12.  At  8%,  interest  compounded  quarterly,  how  much 
will  $850  amount  to  in  1  yr.  10  mo.  20  da.  ? 

13.  What  will  $500  amount  to  in  20  yr.  at  7%,  comp.  int.  ?. 

14.  A  father  at  his  death  left  $12500  for  the  benefit  of 
his  only  son,  14  yr.  8  mo.  12  da.  old,  the  money  to  be  paid 
him  when  he  should  be  21  years  of  age,  with  6%  interest 
compounded  semi-annually.     What  did  he  receive  ? 

amual  Interest. 

588.  Annual  Interest  is  interest  on  the  principal 

and  on  each  year's  interest  remaining  unpaid,  but   so 

computed  as  not  to  increase  the  original  principal. 

It  is  allowed  in  the  case  of  promissory  notes  and  other  contracts 
which  contain  the  words,  "  with  interest  payable  annually,"  or  with 
"  compound  interest. "  In  such  cases,  the  interest  is  not  compounded 
beyond  the  second  year. 


INTEREST.  313 

WRITTEN    EXERCISES. 

589.  1.  Find  the  annual  interest  and  amount  of  $8000 
for  5  yrv  at  6%  per  annum. 

operation.  Analysis.— The  in- 

Int.  of  $8000  for  5  yr.  at  6^=82400.    terest  on  $8000  for  1 

"    "  $480  for  10  yr.  at  6%  =  $288.     J*  f  ^  ^i?™  ""* 
J  '  for  5  yr.  is  $2400. 

$2400  +  $288=$2688,  Annual  int.  The  interest  for  the 

$8000 +  $2688  =  $10688,  Amount.  first  year,   remaining 

unpaid,  draws  interest 
for  4  yr. ;  that  for  the  second  year,  for  3  yr.  ;  that  for  the  third  year, 
for  2  yr.  ;  and  that  for  the  fourth  year,  for  1  yr.,  the  sum  of  which 
is  equal  to  the  interest  of  $480  for  4  yr.  +  3  yr.  +  2  yr.  + 1  yr.  =10  yr. ; 
and  the  interest  of  $480  at  6^  for  10  yr.  is  $288.  Hence  the  total 
amount  of  interest  is  $2400  +  $288,  or  $2688,  and  the  amt.  is  $10688. 

2.  What  is  the  annual  interest  of  $1500  for  4  yr.  at  7$? 

Rule.  —  Compute  the  interest  on  the  principal  for  the 
given  time  and  rate,  to  which  add  the  interest  on  each 
year's  interest  for  the  time  it  has  remained  unpaid. 

To  obtain  the  latter,  when  the  interest  has  remained 
unpaid  for  a  number  of  years,  multiply  the  interest  for 
one  year  by  the  product  of  the  number  of  years  and  half 
that  number  diminished  by  one. 

Thus,  if  the  time  is  9  yr.,  the  Interest  for  1  yr.  should  be  multi- 
plied by  9  x  (9  -  1)  -r-  2,  or  9  x  4  =  36.  Since  the  interest  for 
the  first  year  draws  8  years'  interest,  that  for  the  second  year  7 
years'  interest,  etc.,  and  the  sum  of  the  series  8  +  7  +  6  +  5  +  4  +  3  +  2 
+  lis36. 

3.  What  will  $3500  amt.  to  in  10  yr.,  annual  int.,  at  8%? 

4.  What  is  the  difference  between  the  annual  interest 
and  the  compound  interest  of  $2500  for  6  yr.  at  6%? 

5.  Find  the  amt.  of  $575,  at  8%  annual  int.,  for  9£  yr. 


314  PERCENTAGE. 

6.     $800.  Macon,  June  15,  1872. 

Four  years  after  date,  for  value  received,  I  promise  to  pay 
Robert  E.  Park,  or  order,  eight  hundred  dollars,  with  in- 
terest at  seven  per  cent.,  payable  annually. 

J.  W.  Burke. 

What  amount  is  due  on  this  note  at  maturity,  no  in- 
terest having  been  paid  ? 

PAET1AL    PAYMENTS. 

590.  Partial  Payments  are  payments  in  part  of 
the  amount  of  a  note,  bond,  or  other  obligation. 

591.  Indorsements  are  the  acknowledgment  of 
such  payments,  written  on  the  back  of  the  note,  bond, 
etc.,  stating  the  time  and  amount  of  the  same. 

592.  A  Promissory  Note  is  a  written  promise  to 
pay  a  certain  sum  of  money,  on  demand  or  at  a  specified 
time. 

593.  The  Maker  or  Drawer  of  the  note  is  the 
person  who  signs  it. 

594.  The  Payee  is  the  person  to  whom,  or  to 
whose  order,  the  money  is  paid. 

595.  An  Inilorser  is*  a  person  who,  by  signing 
his  name  on  the  back  of  the  note,  makes  himself  respon- 
sible for  its  payment. 

596.  The  JFace  of  a  note  is  the  sum  of  money  made 
payable  by  the  note. 

597.  A  Negotiable  Note  is  one  made  payable  to 
bearer,  or  to  any  person's  order.  When  so  made,  it  can 
be  sold  or  transferred. 


PAETIAL     PAYMENTS.  315 

WHITTEN    EXERCISES. 

1.     $800.  New  York,  Jan.  1st,  1874. 

One  year  after  date,  I  promise  to  pay  Caleb  Barlow,  or 
order,  eight  hundred  dollars,  for  value  received,  with  in- 
terest.  James  Dunlap. 

Indorsed  as  follows  :  April  1,  1874,  $10  ;  July  1,  1874, 
$35  ;  Nov.  1, 1874,  $100.  What  was  there  due  Jan.  1, 1875  ? 

Analysis. — The  interest  of  $800  for  3  mo.,  from  Jan.  1  to  April  1, 
at  7fe,  is  $14;  am't,  $814.  Since  the  payment  is  less  than  the  in- 
terest, it  cannot  be  deducted  for  a  new  principal  without  com- 
pounding the  int.,  which  is  illegal ;  hence,  find  the  int.  of  $800  to 
the  time  of  the  next  payment,  3  mo.,  which  is  $14,  and  the  amt.  to 
that  time,  $828,  from  which  deduct  the  sum  of  the  two  payments, 
or  $45,  leaving  $783,  a  new  principal.  The  int.  of  $783  for  4  mo., 
to  Nov.  1,  is  $18.27;  amt.,  $801.27;  from  which  deduct  the  third 
payment,  $100,  leaving  $701.27,  the  next  principal,  the  amt.  of 
which  for  2  mo.,  to  Jan.  1,  1874,  is  $709.45,  sum  due  at  that  time. 

Principle. — The  principal  must  not  be  increased  by  the 
addition  of  interest  due  at  the  time  of  any  payment,  so  as 
to  compound  the  interest. 

Upon  this  principle  is  based  the  rule  prescribed  by  the 
Supreme  Court  of  the  United  States  : 

U.  S.  Rule. — I.  Find  the  amount  of  the  given  princi- 
pal to  the  time  of  the  first  payment,  and  if  this  payment 
equals  or  exceeds  the  interest  then  due,  subtract  it  from  the 
amt.  obtained,  and  treat  the  remainder  as  a  new  principal. 

II.  If  the  interest  exceed  the  payment,  find  the  amount 
of  the  same  principal  to  a  time  tuhen  the  sum  of  the  pay- 
ments equals  or  exceeds  the  interest  then  due,  and  subtract 
the  sum  of  the  payments  from  that  amount. 

III.  Proceed  in  the  same  manner  with  the  remaining 
payments. 


316  PEBCENTAGE. 

$500.  Philadelphia,  Feb.  1,  1875. 

2.  Three  months  after  date,  I  promise  to  pay  to  J.  B. 
Lipirincott  &  Co.,  or  order,  jive  hundred  dollars,  with 
interest,  without  defalcation.     Value  received. 

James  Monkoe. 

Indorsed  as  follows  :  May  1, 1875,  $40  ;  Nov.  14, 1875, 
$8;  April  1,  1876,  $18;  May  1,  1876,  $30.  What  was 
due  Sept.  16,  1876  ? 

OPERATION. 

Face  of  note,  or  principal $500.00 

Interest  to  May  1,  1875,  3  mo.,  at  Qf0       .......  7.50 

Amount 507.50 

Payment,  to  be  subtracted 40.00 

2d  principal 467.50 

Int.  of  $467.50  to  Nov.  14,  1875,  6  mo.  13  da.  .    .     $15.04 

Int.  of  $467.50  to  April  1,  1876,  4  mo.  17  da.    .    .      10.67  25.71 

Amount 493.21 

Sum  of  payments,  to  be  subtracted 26.00 

3d  principal 467.21 

Int.  to  May  1,  1876,  1  mo 2.34 

Amount 469.55 

Payment,  to  be  subtracted 30.00 

4th  principal 439.55 

Int.  to  Sept.  16,  1876,  4  mo.  15  da., 9.89 

Amount  due $449.44 

3.  What  was  the  amount  due  October  25,  1873,  upon  a 
note  for  $1500,  dated  New  Orleans,  April  1,  1872,  and 
on  which  the  following  payments  were  endorsed  :  June  5, 
1872,  $300 ;  Oct.  15,  1872,  $37.75;  May  1,  1873,  $97.25 ; 
Aug.  6,  1873,  $495? 


PARTIAL     PAYMENTS.  317 


$700.  Detroit,  Nov.  1,  1873. 

4.  Oth  demand,  1  promise  to  pay  Charles  Smith,  or 
order,  seven  hundred  dollars,  with  interest.  Value  re- 
ceived. Abraham  Isaacs. 

Indorsed  as  follows  :  Dec.  5,  1873,  $75  ;  Jan.  10,  1874, 
$350;  April  11,  1874,  $11.25;  May  15,  1874,  $250. 
What  was  due  Sept.  1,  1874? 


$497^q.  Chicago,  March  15,  1874. 

5.  Three  months  after  date,  we  promise  to  pay  James 
Kelly,  or  order,  four  hundred  and  ninety-seven  -ffo  dollars, 
with  interest.     Value  received. 

Brown,  Nichols  &  Co. 

Indorsed  as  follows  :  Nov.  3,  1874,  $57.50 ;  June  15, 
1875,  $22.25  ;  Aug.  1,  1875,  $125 ;  Sept.  15, 1875,  $175. 
What  was  due  Jan.  1,1876? 

598.  The  following  method  of  computation  is  often 
used  by  merchants  in  the  settlement  of  notes  and  of  in- 
terest accounts  running  a  year  or  less ;  hence  called  the 
Mercantile  Rule: 

I.  Find  the  amount  of  the  note  or  debt  from  its  date 
to  the  time  of  settlement. 

II.  Find  the  amount  of  each  payment  from  its  date 
to  the  time  of  settlement. 

III.  Subtract  the  sum  of  the  amounts  of  payments  from 
the  amount  of  the  note  or  de$t. 

An  accurate  application  of  this  rule  requires  that  the  time  should  be  reduced 
to  days,  and  that  the  interest  should  be  computed  by  the  rale  for  days  (574). 

For  the  Vermont  State  method  of  computation,  and  also  of  assessing  taxes, 
see  pages  491-495. 


318  PERCENTAGE. 

1.  On  a  note  for  $600  at  1%,  dated  Feb.  15,  1874,  were 
the  following  indorsements  :  March  25,  1874,  $150  ;  June 
1, 1874,  $75 ;  Oct.  10,  1874,  $100.    What  was  due*Dec.  31, 

1874? 

OPERATION. 

Am't  of  $600  from  Feb.    15  to  Dec.  31,  319  da.,  $636.71 

"  "  $150  "  Mar.  25  "  "  281  da.,  $158.08 
"  u  $75  "  June  1  "  "  213  da.,  78.06 
"     "  $100     "     Oct.    10  "        '*  82  da.,       101.57        337.71 

Balance  due  Dec.  31,  1874,  $299.00 

2.  A  note  for  $950,  dated  Jan.  25,  1876,  payable  in 
9  mo.,  at  7$  interest,  had  the  following  indorsements  : 
March  2,  1876,  $225 ;  May  5,  1876,  $174.19  ;  June  29, 
1876,  $187.50;  Aug.  1,  1876,  $79.15.  What  was  the 
balance  due  at  the  time  of  its  maturity  ? 

3.  Payments  were  made  on  a  debt  of  $1750,  due  April  5, 
1875,  as  follows*  May  10,  1875,  $190 ;  July  1,  1875, 
$230  ;  Aug.  5,  1875,  $645  ;  Oct.  1,  1875,  $372.  What 
was  due  Dec.  31, 1875,  interest  at  6%  ? 

DISCOUNT. 

599.  Discount  is  a  certain  per  cent  deducted  from 
the  price-list  of  goods,  or  an  allowance  made  for  the  pay- 
ment of  a  debt  or  other  obligation  before  it  is  due. 

600.  The  Present  Worth  of  a  debt  payable  at  a 
future  time  without  interest,  is  such  a  sum  as,  being  put 
at  legal  interest,  will  amount.to  the  debt  when  it  becomes 
due. 

601.  The  True  Discount  is  the  difference  between 
the  whole  debt  and  the  present  worth. 


DISCOUNT.  319 

ORAL     EXERCISES. 

602.  1.  What  is  the  present  worth  of  a  debt  of  $224, 

to  be  paid  in  2  yr.,  at  6%  ? 

Analysis.— Since  in  2  yr.,  at  Qfo,  the  int.  is  .12  of  the  principal, 
the  amt  is  1.12  of  it ;  therefore,  $224,  the  debt,  is  1.12,  or  \%%  of 
the  present  worth,  and  |$#,  or  the  present  worth  itself,  is  $200. 
Or,  since  $1.12  is  the  amt.  of  $1,  $224  is  the  amt.  of  as  many  dol- 
lars as  $1.12  is  contained  times  in  $224,  or  $200.    (578.) 

What  is  the  present  worth 

2.  Of  $315,  due  in  10  mo.,  at  6%  ? 

3.  Of  $570,  due  in  2  yr.,  at  7%  ? 

4.  Of  $408,  due  in  3  mo.,  at  8%  ? 

5.  Of  $51,  due  in  4  mo.,  at  6%  ? 

6.  Of  $440,  due  in  2  yr.,  at  5%  ? 
Find  the  true  discount  at  Q%, 

7.  Of  $1019,  due  in  3  mo.  24  da. 

8.  Of  $102.20,  due  in  4  mo.  12  da. 

9.  Of  $5035,  due  in  1  mo.  12  da. 

WRITTEN    EXERCISES. 

603.  1.  What  are  the  present  worth  and  the  true  dis- 
count, of  $362.95,  payable  in  7  mo.  12  da.,  at  6%  ? 

OPERATION. 

Amt.  of  $1,  for  7  mo.  12  da.,  at  G%  as  $1,037 
$362.95  -j-  $1,037  =  350  times. 
$1  x  350  as  $350,  Present  Worth. 
$362.95  —  $350  as  $12.95,  True  Discount. 

Analysis.— Since  the  amount  of  $1  for  7  mo.  12  da.  at  6%  is 
$1,037  (570),  $362.95  is  the  amount  of  as  many  dollars  as  $1,037 
is  contained  times  in  $362.95,  or  350  times.  Hence  the  present 
worth  is  $350  ;  and  the  true  discount  is  $362.95  —  $350,  or  $12,95. 


320  PERCENTAGE. 

2.  What  is  the  present  worth  of  a  debt  of  $287.75  to  be 
paid  in  3  mo.  18  da.  at  7%  ? 

3.  What  is  the  true  discount  on  a  debt  of  $2202.90  due 
in  8  mo.  12  da.  at  H%  ? 

Rule. — I.  Divide  the  debt  by  the  amount  of  $1  for  the 
given  rate  and  time,  and  the  quotient  is  the  present  worth. 

II.  Subtract  the  present  worth  from  the  debt,  and  the 
remainder  is  the  true  discount. 

Formula. — Present  Worth  =  Debt  ~  Amt.  of%\. 

Hence  the  present  worth  is  the  principal  of  which  the  true  dis- 
count is  the  interest,  and  the  whole  debt  the  amount. 

4.  Bought  a  house  and  lot  for  $19500  cash,  and  sold 
them  for  $22000,  payable  one-fourth  in  cash  and  the  re- 
mainder in  1  yr.  6  mo.  How  much  ready  money  did  I 
gain,  computing  discount  at  6%  ? 

5.  A  merchant  buys  goods  for  $4200  on  4  mo.  credit, 
but  is  offered  a  discount  of  3%  for  cash.  If  money  is 
worth  \%  ft  month,  what  is  the  difference  ? 

6.  Bought  a  bill  of  lumber  amounting  to  $3500,  on 
6  mo.  credit ;  2  months  afterward  paid  on  account  $1500, 
and  1  month  later,  $1000.  Find  the  present  worth  of 
the  balance,  at  the  time  of  the  second  payment,  int.  at  7%. 

7.  A  merchant  holds  two  notes,  one  for  $356.25  due 
Dec.  1,  1875,  and  the  other  for  $497.50,  due  Feb.  1, 1876. 
What  would  be  due  him  in  cash  on  both  notes  Sept.  15, 
1875,  at  6%  ? 

8.  A  bookseller  bought  $300  worth  of  books  at  a  dis- 
count of  33£$  from  list  prices,  and  sold  them  at  the  reg- 
ular retail  price,  on  6  mo.  time.  Money  being  worth  6%, 
what  per  cent,  profit  did  he  make  ? 


DISCOUNT.  321 

9.  A  speculator  bought  230  bales  of  cotton,  each  bale 
containing  470  lb.,  at  llf  cents  a  pound,  on  a  credit  of 
9  mo.  He  at  once  sold  the  cotton  for  $13000  cash,  and 
paid  the  pres.  worth  of  the  debt  at  7$.   What  was  his  gain  ? 

10.  Which  is  the  more  profitable,  to  buy  flour  at  $8.75  a 
barrel  on  6  mo.  credit,  or  at  $8.60  on  2  mo.,  money  being 
worth  7$?     . 

11.  A  person  sold  goods  to  the  amount  of  $3750,  15$ 
payable  in  cash,  25$  in  3  mo.,  20$  in  4  mo.,  and  the  re- 
mainder in  6  mo.  What  ready  money  would  discharge 
the  whole  debt,  money  being  worth  6%  ? 

BANK   DISCOUNT. 
G04.  A  Bank  is  a  corporation  chartered  by  law  for 
the  safe-keeping  and  loaning  of  money,  or  the  issuing  of 
bills  for  circulation  as  money. 

605.  Bank  Bills  or  Notes  are  promissory  notes 

issued  by  banks,  and  payable  on  demand. 

A  bank  which  issues  notes  to  circulate  as  money  is  called  a  Bank 
of  Issue  ;  one  which  lends  money  by  discounting  notes,  a  Bank  of 
Discount ;  and  one  which  takes  charge  of  money  belonging  to  other 
parties,  called  depositors,  a  Savings  Bank,  or  Bank  of  Deposit. 
Some  banks  perform  two  and  others  all  of  these  duties. 

606.  Bank  Discount  is  a  deduction  made  for 
interest  in  advancing  money  upon  a  note  not  due,  or  pay- 
ment by  a  borrower,  in  advance,  of  interest  upon  money 
loaned  to  him.  It  is  equal  to  the  interest  at  the  given 
rate  for  the  given  time  (including  the  days  of  grace)  on 
the  whole  sum  specified  to  be  paid. 

607.  Days  of  Grace  are  the  three  days  allowed 
by  law  for  the  payment  of  a  note  after  the  expiration  of 
the  time  specified  in  the  note.  They  are  counted  in  by 
bankers  in  discounting  notes. 


322  PERCENTAGE. 

608.  The  Maturity  of  a  note  is  the  expiration  of 
the  whole  time,  including  the  days  of  grace. 

609.  The  Term  of  Discount  is  the  time  from  the 
discount  of  a  note  to  its  maturity. 

610.  A  Sank  Chech  is  a  written  order  for  money 
by  a  depositor,  upon  a  bank. 

611.  The  Proceeds,  or  Avails  of  a  note  is  the 
sum  received  for  it  when  discounted,  that  is,  the  face  of 
the  note  less  the  discount. 

612.  A  Protest  is  a  formal  declaration  in  writing, 
made  by  a  Notary-Public,  at  the  request  of  the  holder  of 
a  note,  to  give  legal  notice  to  the  maker  and  the  indorsers 
of  its  non-payment. 

1.  The  failure  to  protest  a  note  on  the  third  day  of  grace  releases 
the  indorsers  from  all  obligation  to  pay  it. 

2.  If  the  third  day  of  grace  or  the  maturity  of  a  note  occurs  on 
Sunday,  or  a  legal  holiday,  it  must  be  paid  on  the  day  previous. 

3.  The  transaction  of  borrowing  money  at  a  bank  is  conducted 
as  follows  :  The  borrower  presents  a  note,  either  made  or  indorsed 
by  himself,  payable  at  a  specified  time,  and  receives  for  it  a  sum 
equal  to  the  face  less  the  interest  for  the  time  it  has  to  run,  in- 
eluding  the  days  of  grace.  A  note  for  discount  at  a  bank  must  be 
made  payable  to  the  order  of  some  person,  by  whom  it  must  be 
indorsed.  When  the  note  bears  interest,  the  discount  is  computed 
on  its  face  plus  the  interest  for  the  time  it  has  to  run. 

613.  Bank  discount  being  simple  interest,  the  follow- 
ing are  corresponding  terms  : 

The  Face  of  the  Note  is  the  principal. 

The  Term  of  Discount  is  the  time. 

The  Bank  Discount  is  the  interest. 

The  Proceeds  is  the  principal  less  the  interest. 


discount.  323 

614.  To  find  the  bank  discount  and  proceeds  oi 
a  note. 

ORAL      EXERCISES. 

1.  What  is  the  bank  discount  on  a  note  for  $2000  due 
in  2  mo.  15  da.  at  6$,  and  the  proceeds  ? 

Analysis. — After  adding  3  da.,  the  time  is  2  mo.  18  da.  ;  the  in- 
terest for  which  at  6%  is  .013  of  the  principal  ;  .013  of  $2000  is  $26, 
the  bank  discount,  and  $2000  —  $26  equals  $1974,  or  the  proceeds. 

What  are  the  bank  discount  and  the  proceeds  of  a  note 

2.  Of  $80  for  5  mo.  27  da.,  at  1%  ? 

3.  Of  $100  for  2  mo.  21  da.,  at  6%  ? 

4.  Of  $200  for  8  mo.  9  da.,  at  1%  ? 

5.  Of  $150  for  4  mo.  21  da.,  at  5%  ? 

6.  Of  $100  for  30  da.,  at  6%  ? 

WRITTEN    EXERCISES. 

615.  1.  Required  the  bank  discount  and  proceeds  of 
a  note  for  $1250  due  in  90  days,  at  1%. 

OPERATION. 

$1250 x. 07  x 93  =  $22  32?  Bank  Discoimt. 

000 

$1250  —  $22.32  =  $1227.68,  Proceeds. 

Analysis.— The  interest  of  $1250  for  93  da.,  at  7%,  reckoning 

365  da.  to  the  year,  is  $22.32,  which  is  the  bank  discount.    If  360  da. 

are  reckoned  to  the  year,  the  bank  disc't  is  $22,604.     Deducting  the 

bank  disc't  from  the  face  of  the  note,  the  remainder  is  the  proceeds. 

Rule. — I.  Compute  the  interest  on  the  face  of  the  note 
(or  if  it  bears  interest,  on  its  amount  at  maturity),  for 
three  days  more  than  the  specified  time,  and  the  result  is 
the  lank  discount. 

II.  Subtract  the  discount  from  the  face  of  the  note,  or 
its  amount  at  maturity,  and  the  remainder  is  the  proceeds. 


324  PERCENTAGE. 

2.  What  is  the  bank  discount,  and  what  is  the  pro- 
ceeds of  a  note  for  $597.50,  due  in  60  da.,  at  $%  ? 

3.  What  will  be  the  proceeds  of  a  note  for  $1615,  due 
in  90  da.  with  interest  at  3%  discounted  at  the  Nassau 
Bank  in  New  York  ? 

4.  Sold  a  farm,  containing  173  A.  95  P.,  for  $62 \  an 
acre,  and  received  payment  as  follows  :  $2000  cash,  and 
the  balance  in  a  note  payable  in  5  mo.  18  da.  at  H%  inter- 
est, which  was  discounted  at  a  bank.  How  much  ready 
money  did  the  farm  bring  ? 

Find  the  date  of  maturity,  the  term  of  discount,  and 
the  proceeds  of  the  following  : 

$957JL3_  Chicago,  July  27,  1875. 

5.  Three  months  after  date,  I  promise  to  pay  to  the 
order  of  D.  L.  Moody,  nine  hundred  fifty-seven  and  -fifc 
dollars,  for  value  received. 

♦Discounted  Aug.  10,  at  %%.       William  Thomson. 

$916^.  San  Francisco,  Feb.  5,  1874. 

6.  Two  months  after  date,  we  jointly  and  severally 
agree  to  pay  C.  H.  Thomas,  or  order,  nine  hundred  six- 
teen and  ^j  dollars  with  interest  at  8%,  •  value  received. 

Discounted  at  Marine  Bank,  James  Barnes. 

Feb.  21,  at  10$.  George  Childs. 

$1315^.  New  York,  May  1, 1875. 

7.  Ninety  days  after  date,  I  promise  to  pay  to  the 
order  of  Ivison,  Blakeman,  Taylor  &  Co.,  one  thousand 
three  hundred  fifteen  and  1%-  dollars,  for  value  received. 

Discounted  May  15,  at  7#.  William  Hewson. 

*  Banks  usually  count  the  actual  number  of  days  in  the  given  time,  and 
865  days  to  the  year. 


discount.  325 

$1250.  Boston,  June  12,  1876. 

8.  Six  months  after  date,  I  promise  to  pay  Knight, 
Adams  &  Co.,  or  order,  twelve  hundred  fifty  dollars,  with 
interest  at  6  per  cent.,  value  received. 

Discounted  at  a  broker's,  Geo.  B.  Damon. 

Nov.  15,  at  6%. 

616.  The  proceeds  and  time  of  a  note  given,  to 
find  the  face. 

ORAL      EXERCISES, 

1.  For  what  sum  must  a  note  be  drawn,  at  2  mo.  15  da., 
at  6$,  so  that  the  proceeds  when  discounted  may  be  $987  ? 

Analysis.— The  bank  discount  for  2  mo.  18  da.  at  6%  is  .013  of 
the  face  of  the  note,  and  the  proceeds  must  therefore  be  1  —  .013, 
or  .987  of  the  face  ;  and  if  .987  of  the  face  is  $987,  the  whole  face 
of  the  note  is  $1000. 

Required  the  face  of  a  note,  so  that  the  proceeds  maybe 

2.  $972,  for  4  mo.  21  da.  at  7%. 

3.  8194,  for  5  mo.  27  da.  at  6%. 

4.  $97.60,  for  3  mo.  15  da.  at  8%. 

5.  $980,  for  4  mo.  21  da.  at  h%. 

6.  $184,  for  9  mo.  15  da.  at  10$. 

WRITTEN     EXERCISES. 

617.  1.  What  must  be  the  face  of  a  note  at  9  mo. 
27  da.,  interest  8$,  so  that  the  proceeds  may  be  $448  ? 

OPERATION. 

The  bank  discount  of  $1  for  10  mo.  at  8%  is  $.06 Gf. 
The  proceeds  of  $1  =  $1  —  $.066|  or  $.933^. 
Hence  $448  -s-  .933£  =  $480,  the  face  of  the  note. 

2.  What  is  the  face  of  a  note  at  30  da.,  the  proceeds  of 
which,  when  discounted  at  bank,  at  7%,  are  $1425  ? 


326  PERCENTAGE. 

Eule. — Divide  the  given  proceeds  by  the  proceeds  of  $1 
for  the  time  and  rate  given ;  the  quotient  is  the  face  of 
the  note. 


Eobmula. — Face  =  Proceeds  ~  (1  —  Bate  x  Time). 

3.  Find  the  face  of  a  3  mo.  note  the  proceeds  of  which, 
discounted  at  2%  a  month,  is  $675. 

4.  The  proceeds  of  a  note  are  $1915.75,  the  time  3  mo., 
and  the  rate  of  interest  7% ;  what  is  the  face  of  the  note  ? 

5.  Bought  merchandise  for  $2250,  cash  ;  for  what  sum 
must  I  draw  my  note  at  3  mo.,  so  as  to  obtain  that  sum 
at  the  bank,  interest  at  7%  ? 

6.  The  avails  of  a  3  months  note,  when  discounted  at 
7\%,  were  $315.23  ;  what  was  the  face  of  the  note  ? 

7.  For  what  sum  must  a  note  dated  «April  5,  for  90  da., 
be  drawn,  so  that  when  discounted  at  7%,  on  April  21, 
the  proceeds  may  be  $650  ? 

8.  For  how  much  must  I  draw  my  note  at  90  da.,  in 
order  that  when  discounted  at  a  bank,  at  7%,  its  avails 
will  pay  for  137f  yd.  of  cloth  at  $2|  a  yard? 

SAVINGS-BANK  ACCOUNTS. 

618.  A  Savings- Bank  is  designed  chiefly  to  ac- 
commodate depositors  of  small  sums  of  money. 

Interest  is  allowed  semi-annually  on  all  sums  that  have  been  ou 
deposit  for  a  certain  time,  if  not  drawn  out  before  the  regular  day 
of  paying  interest— generally  on  the  1st  of  January  and  of  July. 

Savings-banks  generally  allow  interest  only  from  the  commence- 
ment of  each  quarter  ;  but  in  some  banks  money  deposited  pre- 
vious to  the  1st  day  of  any  month  draws  interest  from  that  date  to 
the  day  of  declaring  interest  dividends,  provided  it  has  not  been 
previously  withdrawn. 


DISCO  UKT. 


327 


WRITTEN     EXERCISES. 

619.  1.  A  person  had  on  deposit  Jan.  1,  1874,  $150. 
His  subsequent  deposits  were,  Feb.  3,  $35 ;  March  29, 
$20  ;  April  10,  $43  ;  May  15,  $26.  His  drafts  during  the 
same  time  were,  Jan.  15,  $50 ;  Feb.  27,  $15 ;  April  19, 
$45.    What  interest  was  due  July  1st,  at  6%  ? 

OPERATION. 


Date  of 

Int.  paym'ts. 

Balance 
1st  of  month. 

Smallest  Bal. 
during  mo. 

Interest 
for  1  month. 

SmallcetBal. 
dur'g  Q/rter. 

Interest  for 
1  Quarter. 

Jan.    1 

$150 

Feb.    1 

100 

$100 

$.50 

Mar.    1 

120 

100 

.50 

Apr.   1 

140 

120 

.60 

$100 

$1.50 

May    1 

138 

138 

.69 

June   1 

164 

138 

.69 

July  1 

164 

164 

.82 

138 

2.07 

$3.80 


$3.57 


Balance  due,  with  int.  by  monthly  periods,  $167.80. 
"      "     "  quarterly      "       $167.57. 

Analysis. — At  the  end  of  January,  the  balance  due  is  $100,  which 
having  been  on  deposit  for  the  month,  draws  interest  for  1  mo.  ;  at 
the  end  of  February,  the  balance  is  $120  ;  but  the  smallest  balance 
during  the  month  is  $100  ;  hence  interest  is  allowed  only  on  that 
sum.  The  same  principle  applies  to  the  other  balances.  If  only 
quarterly  periods  of  interest  are  allowed,  the  interest  is  calculated 
at  the  end  of  each  quarter  on  the  smallest  balance  during  the  quar. 
ter,  or,  in  this  case,  on  $100,  April  1,  and  $138,  July  1. 

2.  Find  the  balance,  due  July  1,  on  the  following 
account :  Deposits,  Jan.  15,  $175  ;  April  10,  $60  ;  May  31, 
$110.  Drafts,  March  5,  $75 ;  May  1,  $35  ;  June  10,  $50. 
Interest  at  6%,  from  the  1st  day  of  each  month. 


328 


PERCENTAGE 


3.  A  person  deposits  in  a  savings-bank  the  following 
sums :  Jan.  1,  $350 ;  Feb.  5,  $150 ;  March  15,  $75  ; 
May  10,  $30  ;  June  15,  $100.  During  the  same  time  he 
draws,  Jan.  15,  $150 ;  Feb.  10,  $200 ;  March  31,  $50 ; 
June  1,  $75.  What  interest  at  6$,  payable  from  the  1st 
of  each  month,  must  be  added  to  the  account  July  1  ? 

4.  Balance  the  following,  Jan.  1,  1875  :  Balance  due  to 
Margaret  Brown,  July  1,  1874,  $275.  Deposits  received 
as  follows  :  Aug.  1,  $125 ;  Sept.  15,  $57 ;  Oct.  10,  $350. 
Drafts  paid :  July  15,  $100  ;  Sept.  1,  $150 ;  Nov.  15, 
$68  ;  Dec.  15,  $125.  Interest  at  6%,  from  the  1st  of  each 
quarter,  July  1  and  Oct.  1. 

Rule. — At  the  end  of  each  term  compute  the  interest  for 
the  term  on  the  smallest  balance  on  deposit  at  any  time 
during  the  quarter  ;  and  at  the  end  of  each  period  of  six 
months  add  to  the  balance  of  principal  the  whole  amount  of 
interest  due,  and  the  sum  will  be  the  principal  at  the  com- 
mencement of  the  next  six  months, 

5.  How  much  was  due  Jan.  1,  1876,  on  the  following 
account,  allowing  interest,  computed  from  the  1st  of  each 
quarter,  Jan.  1  to  July  1,  at  6%  per  annum  ? 

Br.  Greenwich  Savings  Bank,  in  acct.  with  Mary  Williams.  (Jr. 


1874. 

1874. 

Jan.     1 

To  Cash  . 

$136 

00 

Sept.  15 

By  Check 

$75 

00 

Mar.  17 

<<         n 

25 

00 

1875. 

Aug.    1 

<<         t< 

87 

50 

Jan.  20 

K                   (( 

37 

50 

1875. 

Mar.    8 

<t                  <( 

50 

00 

June  11 

n        n 

150 

00 

Nov.  17 

<<         i< 

72 

00 

REVIEW 


329 


620. 


SYNOPSIS  FOR  REVIEW. 


r  1.  Defs. 

2. 
3. 
4. 


10.  Interest.  ■{  5. 
6. 


8.  Problems. 


11.  Compound 
Interest 

12.  Annual 
Interest.  /  2 


is 


1.  Interest.  2.  Principal.  3.  itote. 

4.  Jratf.    5.  Ze^aJ  .Zn£.    G.  Usury. 

Corresponding  Elements. 

1.  Principle.     2.  Rule,  I,  II,  III. 

Relations  between  Time  ] 

and  Interest.  j    '     '       ' 

569.    Rule,  I,  II,  III,  IV. 

or?  nr  xi    j   j  1-  Principles,  1,  2. 
6%  Method. -J  3    Rule/ 

Accurate  Interest.     Rule. 

576.  1.  Bute.  2.  Formula. 
578.  1.  Rule.  2.  Formula. 
580.  1.  ifote.  2.  Formula. 
(,  582.  l.ifwfe.  2.  Formula. 
Definitions — Compound  Interest. 
Rule,  I,  II,  III. 
Definitions. 
Rule,  I,  II. 

f  1.  Part.  Pay'ts.    2.  Indorserrits. 
3.  Promissory  Note.    4.  Maker 

1.  Defs.  <       #r  Drawer.    5.  Payee.    6.  7/i 
dorse r.     7.    Face  of  a  Note. 

[      8.  Negotiable  Note. 

2.  Principle.  U.  S.  Rule.  I.  II.  Merc.  Rule, 
j  1.  Discount.  2.  Present  Worth. 
(  3.  2>«e  Discount. 

2.  Rule,  I,  II. 

f  1.  Bank.    2.  Bank  Bills  or  Notes. 
3.  i?aw&  Discount.    4.  .Days  0/ 
1    Defs   J       Grace.    5.  Maturity  of  Note. 
C.  jT<?rm  of  Discount.    7.  ItawA; 
C%erA;.     8.  Proceeds  or  Avails, 
9.  Protest. 
Corresponding  Terms. 
614.     Rule,  I,  II. 
616.    Rule 
16.  Savings-Bank  Accounts— Rule. 


13.  Partial 
Payments. 


14.  Discount. 


15.  Bank  Dis- 
count. 


621.  A  Corporation  is  an  association  of  indi- 
viduals authorized  by  law  to  transact  business  as  a  single 
person. 

622.  A  Charter  is  the  legal  act  of  incorporation 
defining  the  powers  and  obligations  of  the  body  incor- 
porated. 

623.  The  Capital  Stock  of  a  corporation  is  the 
capital  or  money  contributed,  or  subscribed  to  carry  on 
the  business  of  the  company. 

624.  Certificates  of  Stock  or  Scrip  are   the 

papers  or  documents  issued  by  a  corporation,  specifying 
the  number  of  shares  of  the  joint  capital  which  the 
holders  own. 

625.  A  Share  is  one  of  the  equal  parts  into  which 
capital  stock  is  divided. 

The  value  of  a  share  in  the  original  contribution  of  capital  varies 
in  different  companies.  In  bank,  insurance,  and  railroad  compa- 
nies, it  is  usually  $100. 

626.  Stocks  is  a  general  term  applied  to  shares  of 
stock  of  various  kinds,  Government  and  State  bonds,  etc. 

Stockholders  are  the  owners  of  stock,  either  by  original  title  or  by 
subsequent  purchase.     The  stockholders  constitute  the  company. 

627.  The  Par  Value  of  stock  is  the  sum  for  which 
the  scrip  or  certificate  was  issued. 

628.  The  Market  Value  of  stock  is  the  sum  for 

which  it  can  be  sold. 


STOCKS.  331 

Stock  is  at  par  when  it  can  be  sold  for  its  original  or  face  value, 
or  100% ;  it  is  above  par,  or  at  a  premium,  when  it  will  bring  more 
than  its  face  value  ;  and  it  is  below  par,  or  at  a  discount,  when  it 
sells  for  less  than  its  face  value.  Thus,  when  stock  is  at  par,  it  is 
quoted  at  100  ;  when  it  is  5%  above  par,  at  105  ;  and  when  it  is  5% 
below  par,  at  95. 

639.  Premium,  Discount,  and  Brokerage 

are  each  a  percentage  computed  upon  the  par  value  of  the 
stock  as  the  base. 

630.  A  Stock  Broker  is  a  person  who  buys  and 
sells  stocks,  either  for  himself,  or  as  the  agent  of  another. 

631.  Stock-jobbing  is  the  buying  and  selling  of 
stocks  with  the  view  to  realize  gain  from  their  rise  and 
fall  in  the  market. 

633.  An  Installment  is  a  portion  of  the  capital 
stock  required  of  the  stockholders  as  a  payment  on  their 
subscription. 

633.  An  Assessment  is  a  sum  required  of  stock- 
holders, to  meet  the  losses,  or  to  pay  the  business  expenses 
of  the  company. 

634.  A  Dividend  is  a  sum  paid  to  the  stockholders 

from  the  profits  of  the  business. 

Dividends  and  assessments  are  a  percentage  computed  upon  the 
par  value  of  the  stock  as  the  base. 

635.  Net  Earnings  are  the  moneys  left  from  the 
profits  of  a  business  after  paying  expenses,  losses,  and  the 
interest  upon  the  bonds. 

636.  A  Bond  is  a  written  instrument  securing  the 

payment  of  a  sum  of  money  at  or  before  a  specified  time. 

The  principal  bonds  dealt  in  by  brokers  are  Government,  State, 
City,  and  Railroad  bonds. 


332  PERCENTAGE. 

637.  U.  S.  Bonds  are  of  two  kinds  ;  viz.,  those 
which  are  payable  at  a  fixed  date,  and  those  which,  while 
payable  at  a  fixed  date,  may  be  paid  at  an  earlier  specified 
time,  as  the  Government  may  elect. 

1.  The  former  are  quoted  in  commercial  transactions  by  the  rate 
of  interest  which  they  bear  ;  thus,  United  States  bonds  bearing  6  % 
interest  are  quoted  TJ.  8.  6'a.  The  latter  are  quoted  in  commercial 
transactions  by  a  combination  of  the  two  dates  ;  thus,  TJ.  8.  5-20's, 
or  TJ.  8.  6'8  5-20,  means  bonds  of  U.  S.  bearing  6%  interest,  and  pay- 
able at  any  time  from  5  to  20  years,  as  the  Government  may  choose. 

2.  When  it  is  necessary  to  distinguish  different  issues  bearing  the 
same  rate  of  interest,  the  year  at  which  they  become  due  is  also 
mentioned  ;  thus,  TJ.  8.  5's  of  '71;  TJ.  8.  5's  of  '74;  TJ.  8.  6%  5-20, 
of  84;   TJ.  8.  G'8,5-20,  of  85. 

3.  The  5-20's  were  issued  in  1862,  '64,  '65,  '67,  and  70.  They 
bear  interest  at  6%,  paid  semi-annually  in  gold,  except  the  issue  of 
1870,  called  5's  of  '81,  which  bear  int.  at  5%,  paid  quarterly  in  gold. 

4.  Bonds  issued  by  States,  cities,  etc.,  are  quoted  in  a  similar 
manner.  Thus,  8.  G.  6's  are  bonds  bearing  6%  interest,  issued  by 
the  State  of  South  Carolina. 

638.  A  Coupon  is  a  certificate  of  interest  attached 
to  a  bond,  to  be  cut  off  and  presented  for  payment  when 
the  interest  is  due. 

639.  Currency  is  a  term  used  to  denote  the  circu- 
lating medium  employed  as  a  substitute  for  gold  and 
silver.  It  consists,  at  present,  in  the  United  States,  of 
TJ.  S.  Legal-tender  Notes,  or  "Greenbacks,"  and  the 
Bills  issued  by  the  Nat.  Banks,  and  secured  by  TJ.  S.  Bonds. 

If  from  any  cause  the  paper  medium  depreciates  in  value,  gold 
becomes  an  object  of  investment,  the  same  as  stocks.  Gold  being 
of  fixed  standard  value,  its  fluctuations  in  price  indicate  changes 
in  the  value  of  the  currency.  Hence,  when  gold  is  said  to  be  at 
a  premium,  currency  is  virtually  below  par,  or  at  a  discount. 


stocks.  332 

ORAL     EXERCISES, 

640.  1.  Find  the  cost  of  100  shares  of  Chicago  and 

Rock  Island  Railroad  stock  at  90 ;  brokerage  \%. 

Analysis. — Since  the  cost  of  one  share  is  90%  of  $100,  or  $90, 
the  cost  of  100  shares  is  100  times  $90,  or  $9000,  to  which  add  the 
brokerage,  \%  of  $10000,  or  $12|,  and  the  sum  $9012^,  is  the  entire 
cost  of  the  stock. 

2.  What  cost  50  shares  of  N.  Y.  Central  R.  R.  Stock, 
at  par  ;  brokerage,  \%  ? 

3.  Find  the  cost  of  10  shares  of  Bank  Stock  at  104 ; 
brokerage  \%. 

4.  What  is  the  cost  of  $2000  U.  S.  G's  5-20,  at  112 ; 
brokerage  \%  ? 

641.  1.  A  broker  has  $5010  to  invest  in  bank  stock  at 
25%  premium  ;  how  many  shares  can  he  buy,  charging  \% 
for  brokerage  ? 

Analysis. — Since  the  stock  sells  at  25%  premium,  each  share 
with  brokerage  will  cost  $125^  ;  hence  he  can  buy  as  many  shares 
as  $125|  are  contained  times  in  $5010,  or  40  shares. 

2.  A  speculator  invested  $52000  in  Ohio  and  Missis- 
sippi R.  R.  stock  at  25|,  allowing  \%  brokerage ;  how 
many  shares  did  he  buy  ? 

3.  If  I  invest  $2350  in  U.  S.  6's,  '81,  at  117f,  broker- 
age \%,  how  many  $1000  bonds  do  I  receive  ? 

642.  1.  A  man  bought  a  number  of  shares  of  mining 
stock  at  60,  and  sold  the  same  at  68,  and  gained  $800 
by  the  transaction.     How  many  shares  did  he  buy  ? 

Analysis.— Since  he  bought  at  G0%  and  sold  at  68%,  he  gained 
8%  of  the  par  value  ;  hence  $800  is  8%  of  $10000,  the  par  value, 
and  the  number  of  shares  at  $100  each  is  100. 


334  PERCENTAGE. 

2.  Bought  E.  E.  stock  at  90,  and  sold  at  par,  gaining 
$1000.     Eequired  the  number  of  shares. 

3.  I  purchased  stock  at  110  and  sold  at  98,  losing 
$1200.     How  many  shares  did  I  buy  ? 

4.  A  broker  bought  some  stock  at  par,  and  sold  it 
at  95,  losing  $2000.     How  many  shares  did  he  buy  ? 

643.  1.  What  sum  must  be  invested  in  California  7's> 
at  110,  to  obtain  therefrom  an  annual  income  of  $1400 ? 

Analysis. — Since  the  annual  income  is  $7  on  each  share,  the 
number  of  shares  must  be  equal  to  $1400  -i-  $7,  or  200  shares  ;  and 
200  shares  at  $110  amount  to  $22000,  the  required  investment. 

2.  What  sum  must  I  invest  in  stock  at  115,  paying 
10$  yearly  dividends,  to  realize  an  income  of  $2000  ? 

3.  What  sum  must  be  invested  in  N.  Y.  Ts  at  103|, 
in  order  to  receive  therefrom  an  annual  income  of  $2100  ? 

644.  1.  What  per  cent,  does  money  yield  which  is 
invested  in  8%  stock  at  120  ? 

Analysis.— Since  each  share  costs  $120,  and  pays  $8  income,  the 
per  cent,  will  be  Tf^,  or  T^  of  100%,  equal  to  6f  %. 

2.  What  per  cent,  does  stock  yield  when  bought  at 
90,  paying  6%  dividends  ?   When  bought  at  75  ?  At  120  ? 

3.  What  per  cent,  of  interest  does  stock  yield,  which 
pays  5%  semi-annual  dividends,  if  bought  at  150  ?  At 
140?    At  120? 

645.  1.  What  should  be  paid  for  stock  yielding  6% 

dividends,  in  order  to  realize  an  annual  interest  on  the 

investment  of  8%  ? 

Analysis. — Since  the  annual  dividend  on  each  share  is  $6,  this 
must  be  8%  of  the  sum  required  ;  and  if  8%  is  $6,  1%  is  $f,  and 
100%  is  $75.     Hence  the  stock  must  be  bought  for  75. 


STOCKS.  335 

2.  For  what  must  stock  that  pays  7%  dividends  be 
bought  to  realize  10%  interest  ?     9%  ?    8%  ? 

3.  For  what  should  Missouri  6's  be  bought  to  pay  b% 
interest?    5$%?    6^%?    8%? 

646.  1.  How  much  currency  can  be  bought  for  $500 

in  gold,  when  the  latter  is  at  a  premium  of  10$  ? 

Analysis. — Since  $1  in  gold  is  worth  $1.10  in  currency,  $500  in 
gold  is  worth  500  times  $1.10,  or  $550.     Hence,  etc. 

2.  How  much  currency  can  be  bought  for  $200  in 
gold,  when  the  latter  is  at  a  premium  of  9%  ? 

3.  What  is  $1000  in  gold  worth  in  currency,  when 
the  former  is  at  a  premium  of  12$  ?   Of9|%?    OflOJ$? 

647.  1.  How  much  gold  can  be  bought  for  $440  in 

currency,  when  the  former  is  at  a  premium  of  10$  ? 

Analysis.— Since  $1  in  gold  is  worth  $1.10  in  currency,  $440 
will  buy  as  many  dollars  in  gold  as  $1.10  is  contained  times  in 
$440,  or  $400  in  gold.    Hence,  etc. 

2.  How  much  gold  selling  at  9%  premium  will  $1090 
in  currency  buy  ?     $218  ?     $G54  ? 

3.  How  much  gold  at  11$  premium  will  $444  buy  ? 

WRITTEN     EXERCISES. 

648.  Find  the  cost 

1.  Of  220  shares  of  bank  stock,  the  market  value  of 

which  is  103f,  brokerage  \%. 

Operation.— (103f%  +i%)  of  $100  =  $104,  cost  of  1  share. 
$104  x  220  =  $22880,  cost  of  220  shares.    (640.) 

Formula. — Entire  Cost  =  {Market  Value  of  1  Share 
+  Brokerage)  x  No.  of  Shares, 

2.  Find  the  cost  of  350  shares  of  Western  Union  Tele* 
graph  stock,  market  value  9  7 J,  brokerage  \%. 


336  PERCENTAGE. 

3.  A  broker  bought  for  me  15  one-thousand-dollar  U.  S. 
5-20  bonds  at  112  J,  brokerage  \%.    What  was  their  cost  ? 

4.  My  broker  sells  for  me  125  shares  of  stock  at  127$. 
What  should  I  receive,  the  brokerage  being  \%  ? 

649.  Find  the  number  of  shares 

1.  Of  bank  stock  at  105,  that  can  be  bought  for  $25260, 
including  brokerage  at  \%  ? 

Operation.— (105%  +£%)  of  $100  =  $105£,  cost  of  1  share. 
$25260  -*-  $105 J  =  240,  No.  of  shares.    (64 1 .) 

Formula. — No,  of  Shares  =  Investment  -f-  Cost  of  1 
Share. 

2.  How  many  shares  of  N.  J.  Central  E.  R.  stock  at 
107|,  brokerage  \%,  can  be  bought  for  $27000  ? 

3.  How  many  shares  of  Mo.  6's  at  97f ,  brokerage  \%, 
will  $21560  purchase  ? 

4.  Bought  Pacific  Mail  at  29£,  and  sold  at  31J,  paying 
\%  brokerage  each  way.     How  many  shares  will  gain  $330  ? 

Operation.— (31} %  —W>\%)  —  \%  =H%,  gain. 
$330  -4-  $1.50  =  220,  No.  of  shares.     (642.) 

Formula. — No.  of  Shares  =  Whole  Gain  or  Loss  -j- 
Gain  or  Loss  per  Share. 

5.  How  many  shares  of  stock  bought  at  97 \  and  sold 
at  102J,  brokerage  \%  each  way,  will  gain  $990  ? 

6.  Lost  $1680  by  selling  N.  Y.  Central  at  101  that  cost 
104.  Brokerage  being  \%  each  way,  how  many  shares  did 
I  sell  ? 

7.  How  many  shares  of  the  Bank  of  Commerce  bought 
at  110£  and  sold  at  116j,  brokerage  \%  on  the  purchase 
and  the  sale,  will  gain  $1200  ? 


STOCKS.  337 

650.  Find  the  amount  of  investment 
1.  In  U.  S.  5's,  of  '81,  at  111,  so  as  to  realize  therefrom 
an  annual  income  of 


Operation. — $2500 -f- $5,  income  on  1  share  =  500,  No.  of  shares. 
$111,  price  of  1  share  x  500  =a  $55500,  investment.  (643.) 

Formula.—  Investment  =  Price  of  1  Share  x  No.  of 
Shares. 

2.  "What  sum  must  be  invested  in  Tennessee  6?s  at  85, 
to  yield  an  annual  income  of  $1800  ? 

3.  How  much  money  must  be  invested  in  any  stock  at 
105J,  which  pays  5%  semi-annual  dividends,  to  realize  an 
annual  income  of  $2000? 

4.  What  sum  invested  in  stock  at  $63  per  share,  will 
yield  an  income  of  $550,  the  par  value  of  each  share  being 
$50,  and  the  stock  paying  10%  annual  dividends  ? 

651.  Find  the  rate  per  cent,  of  income,  realized 

1.  From  bonds  paying  8%  interest,  bought  at  110. 

Operation. — $8,  interest  per  share  -f-  $110,  cost  per  share  = 
.07^,  or  7^%.    (644.) 

Formula.—  Rate  %  of  Income  ='  Interest  per  Share  -f- 
Cost  per  Share. 

2.  If  stock  paying  10%  dividends  is  at  a  premium  of 
12£%,  what  per  cent,  of  income  will  be  realized  on  an  in- 
vestment in  it  ? 

3.  Which  will  yield  the  better  income,  8%  bonds  at  110, 
or  5's  at  75  ? 

4.  Which  is  the  more  profitable,  and  how  much,  to  buy 
tfew  York  7's  at  105,  or  6  per  cent,  bonds  at  84? 


338  PERCENTAGE. 

5.  What  per  cent,  of  income  does  stock  paying  10$ 
dividends  yield,  if  bought  at  106  ? 

6.  What  per  cent,  will  stock  which  pays  5%  dividends 
yield,  if  bought  at  a  discount  of  15%  ? 

7.  What  rate  per  cent,  of  income  shall  I  receive,  if  I 
buy  U.  S.  5's  at  a  premium  of  10$,  and  receive  payment 

,  at  par  in  15  years  ?  JjL%QJ~-  I  ,  03   $] 

652.  Find  at  what  price  stock  must  he  bought 

1.  That  pays  6%  dividends,  so  as  to  realize  an  income 
of  %%  on  the  investment. 

Operation.— .06  -^  .075  -  .80  or  80%,  price  of  stock.    (645.) 
Formula. — Price  of  Stock  ==  Dividend  -~  Rate  of  In- 
come. 

2.  What  must  be  paid  for  5%  bonds,  that  the  invest- 
ment  may  yield  8%  ? 

3.  How  much  premium  may  be  paid  on  stock  that  pays 
10$  dividends,  so  as  to  realize  7|$  on  the  investment  ? 

4.  What  must  I  pay  for  Government  5's  of  '81,  that  my 
investment  may  yield  7$  ? 

5.  At  what  price  must  stock,  of  the  par  value  of  $50  a 
.  share,  and  that  pays  G$  dividends,  be  bought,  to  yield  an 

income  of  1±%  ? 

6.  At  what  price  must  C$  stock  be  bought,  to  pay  as 
good  an  income  as  8$  stock  bought  at  par  ?    As  9$  stock  ? 

653.  Find  the  value  in  currency, 

1.  Of  $3750  in  gold,  quoted  at  110J, 
Operation.— $1.10i  x  3750= $4143. 75,  value  in  currency.  (640.) 
Formula. — Total  Value  in  Currency  =  Value  of  %\  in 
Currency  x  No.  of  Dollars  in  Gold. 


stocks.  339 

2.  Find  the  value  of  $4975  in  gold,  at  a  premium  of 

3.  What  is  the  semi-annual  interest  of  $8000  6%  gold- 
bearing  bonds  worth  in  currency,  when  gold  is  at  lllf  ? 

4.  A  merchant  bought  a  bill  of  goods,  for  which  he 
was  to  pay  87000  in  currency,  or  $$625  in  gold.  Gold 
being  at  109-f,  which  is  the  better  proposition,  and  how 
much  in  currency  ? 

654.  Find  the  value  in  gold, 

1.  Of  $2150  in  currency,  when  gold  is  at  a  premium  of 

m%. 

Operation.— $2150  -f- 1.105  =  $1945.70,  value  in  gold.     (647.) 
Formula. — Total  Value  in  Gold  =  Amt.  of  Currency 
-r-  (1  4-  Premium), 

2.  What  is  $4500  in  currency  worth  in  gold,  when  the 
latter  is  at  a  premium  of  \%\$  ?    At  \\\%  ?    At  §\%  ? 

j  3.  How  much  money  niust  be  invested  in  U.  S.  6's  at 
111,  when  gold  is  quoted  at  110 J,  in  order  to  obtain  a 
semi-annual  income  of  $2210  in  currency? 

4.  The  Mechanics  Bank  of  New  York  having  $109737.50 
to  distribute  to  the  stockholders,  declares  a  dividend  of 
6 \c/0  ;  what  is  the  amount  of  its  capital  ? 

5.  A  man  owns  a  house  which  rents  for  $1450,  and  the 
tax  on  which  is  %\%  on  a  valuation  of  $8500.  He  sells 
for  $15300,  and  invests  in  stock  at  90,  that  pays  7%  divi- 
dends. Is  his  yearly  income  increased  or  diminished, 
and  how  much  ? 

6.  If  I  have  $36500  to  invest,  and  can  buy  N.  Y.  Cen- 
tral 6's  at  85,  or  N.  Y.  Central  7's  at  95,  how  much  more 
profitable  will  the  latter  be  than  the  former  ? 


kn 


6 


340  PEECENTAGE. 

6/  >■ 

7.  Which  is  the  better  investment,  a  mortgage  for  3  yr. 
of  $5000,  paying  7%  interest,  and  purchased  at  a  discount 
of  5%,  or  50  shares  of  stock  at  95,  paying  8%  dividends, 
and  sold  at  the  expiration  of  3  years  at  98  ? 

8.  Henry  Ivison,  through  his  broker,  invested  a  certain 
sum  of  money  in  New  York  State  6's  at  107J,  and  twice 
as  much  in  U.  S.  5's,  of  '81,  at  98J,  brokerage  in  each 
case  \%.  The  annual  income  from  both  investments  was 
$3348.     How  much  did  he  invest  in  each  kind  of  stock  ? 

9.  A  gentleman  invested  $12480  current  funds  in 
IT.  S.  5-20's  of  '85,  at  104.  What  will  be  his  annual 
income  in  currency  when  gold  is  110? 

IISUEASCE. 

655.  Insurance  is  a  contract  of  indemnity  against 
loss  or  damage.  It  is  of  two  kinds  :  insurance  on  prop- 
erty, and  insurance  on  life. 

656.  The  Insurer  or  Underwriter  is  the  party 
who  takes  the  risk  or  makes  the  contract. 

657.  The  Policy  is  the  written  contract  between  the 
parties. 

658.  The  Premium  is  the  sum  paid  for  insurance, 
and  is  a  certain  per  cent.  X)i  the  sum  insured. 

651).  Insurance  business  is  generally  conducted  by  Companies, 
which  are  either  Joint-stock  Companies,  or  Mutual  Companies. 

A  Stock  Insurance  Company  is  one  in  which  the  capi. 
tal  is  owned  by  individuals  called  stockholders.  They  alone  share 
the  profits,  and  are  liable  for  the  losses. 

A  Mutual  Insurance  Company  is  one  in  which  the  profits 
and  losses  are  divided  among  those  who  are  insured. 

Some  companies  are  conducted  upon  the  Stock  and  Mutual  plans 
combined,  and  are  called  Mixed  Companies. 


INSURANCE.  341 

Insurance  on  property  is  principally  of  two  kinds:  Fire 
Insurance,  and  Marine  and  Inland  Insurance. 

660.  Fire  Insurance  is  indemnity  for  loss  of 
property  by  fire. 

661.  Marine  and  Inland  Insurance  is  in- 
demnity for  loss  of  vessel  or  cargo,  by  casualties  of  navi- 
gation on  the  ocean,  or  on  inland  waters. 

Transit  Insurance  refers  to  risks  of  transportation  by  land  only, 
or  partly  by  land  and  partly  by  water.  The  same  policy  may  cover 
both  Marine  and  Transit  Insurance. 

Stock  Insurance  is  indemnity  for  the  loss  of  cattle,  horses,  etc. 
Most  insurance  companies  will  not  take  risks  to  exceed  two-thirds 
or  three-fourths  the  appraised  value  of  tlie  property  insured. 

When  only  a  part  of  the  property  insured  is  destroyed  or  dam- 
aged, the  insurers  are  required  to  pay  only  the  estimated  loss  ;  and 
sometimes  the  claim  is  adjusted  by  repairing  or  replacing  the 
property,  instead  of  paying  the  amount  claimed. 

662.  The  operations  are  based  on  the  principles  of 
Percentage,  the  corresponding  terms  being  as  follows  : 

1.  The  Hase  is  the  amount  of  insurance. 

2.  The  Mate  is  the  per  cent,  of  premium. 

3.  The  Percentage  is  the  premium. 

ORAL      EXERCISES. 

663.  1.  How  much  must  be  paid  for  insuring  a  house 
and  furniture  for  $4000,  at  \\%  premium  ? 

Analysis. — Since  the  premium  is  1^%,  or  -fa,  equal  to  ¥^  of 
the  sum  insured,  the  premium  on  $4000  will  be  ^  of  $4000,  or 
$50.    Hence,  etc.     (510.) 

2.  What  will  be  the  annual  premium  of  insurance,  at 
£ %,  on  a  building  valued  at  $8000  ? 


342  PERCENTAGE. 

3.  What  will  be  the  cost  of  insuring  a  quantity  of  flour, 
valued  at  $1500,  at  \%  ? 

4.  What  must  be  paid  for  insuring  a  case  of  merchan- 
dise, worth  $640,  at  2\%  ? 

5.  A  man  owns  f  of  a  boat-load  of  corn  valued  at 
$1800,  and  insures  his  interest  at  \\%.  What  premium 
does  he  pay  ? 

6.  Paid  $6  for  insuring  $300  ;  what  was  the  rate  ? 

Analysis. — Since  the  premium  on  $300  is  $6,  the  premium  on 
$1  is  ¥£¥  of  $6>  or  $-03>  e1ual  to  2% .     Hence,  etc.    (513.) 

7.  Paid  $12  for  an  insurance  of  $800  ;  find  the  rate. 

8.  Paid  $24  for  an  insurance  of  $1000  ;  find  the  rate. 

9.  At  2%,  what  amount  of  insurance  can  be  obtained 
for  $30  premium  ? 

Analysis. — Since  2  %  is  T|^  or  -£$  of  the  amount  insured,  $30,  the 
given  premium,  is  -^  of  the  amount  insured ;  and  $30  is  ^  of  50 
times  $30,  or  $1500.     Hence,  etc.    (516.) 

What  amount  of  insurance  can  be  obtained, 

10.  On  a  house,  for  $75,  at  3%  premium  ? 

11.  On  a  boat'  load  of  flour,  for  $150,  at  \%  ? 

12.  On  a  car  load  of  horses,  for  $90,  at  ty%  ? 

13.  On  a  store  and  its  contents,  for  $105,  at  \\%  ? 

WRITTEN    EXERCISES. 

664.  Find  the  Premium 

1.  For  insuring  a  building  for  $14500,  at  1\%. 
Operation.— $14500  x  .015  =  $217.50.    (512.) 
Formula. — Premium  =  Amount  Insured  x  Rate, 

Find  the  premium  for  insuring 

2.  A  house  valued  at  ^700,  at  %%. 

3.  Merchandise  for  $2750,  at  \%* 


INSURANCE.  343 

4.  A  fishing  craft,  for  815000,  at  \\%. 

5.  If  I  take  a  risk  of  $25000,  at  \\%,  and  re-insnre  \ 
of  it  at  %\%,  what  is  my  balance  of  the  premium  ? 

665.  Find  the  Rate  of  Insurance, 

1.  If  $36  is  paid  for  an  insurance  of  $2400. 
Operation.— $36  -r-  $2400  =  .015,  or  \\%.    (515.) 
Formula. — Rate    of  Insurance  =  Premium  —  Sum 

Insured. 

What  is  the  rate  of  insurance, 

2.  If  $280  is  paid  for  an  insurance  of  $16000  ? 

3.  If  $4.30  is  paid  for  an  insurance  of  $860  ? 

4.  A  tea  merchant  gets  his  vessel  insured  for  $20000 
in  the  Royal  Company,  at  f%  and  for  $30000  in  the 
Globe  Company,  at  \%.  What  rate  of  premium  does  he 
pay  on  the  whole  insurance  ? 

666.  To  find  the  Amount  of  Insurance. 

1.  A  speculator  paid  $262.50  for  the  insurance  of  a 
cargo  of  corn,  at  \\%.  For  what  amount  was  the  corn 
insured  ? 

Operation.— $262.50  -^  .015  =  $17500,  the  sum  insured.   (518.) 
Formula. — Sum  Insured  ■=.  Premium  ~-  Rate. 

2.  If  it  cost  $93. 50  to  insure  a  store  for  one-half  of  its 
value,  at  \\%,  what  is  the  store  worth  ? 

3.  Paid  $245  insurance  at  4f  $  on  a  shipment  of  pork, 
to  cover  \  of  its  value.    What  was  its  total  value  ? 

4.  A  merchant  shipped  a  cargo  of  flour  worth  $3597, 
from  New  York  to  Liverpool.  For  what  must  he  insure 
it  at  3\%,  to  cover  the  value  of  the  flour  and  premium  ? 

Operation.— $3597  -f-  (1  -  .03^)  or  .9675  =  $3717.829.    (520.) 


344  PERCENTAGE. 

5.  An  underwriter  agrees  to  insure  some  property  for 
enough  more  than  its  value  to  cover  the  premium,  at  the 
rate  of  26  cents  per  $100.  If  the  property  is  worth 
$22163,  what  should  be  the  amount  of  the  policy  ? 

6.  For  what  sum  must  a  policy  he  issued  to  insure  a 
dwelling-house,  valued  at  $35000;  at  \%,  a  carriage-house 
worth  $9500,  at  \%  and  furniture  worth  $4500",  at  f% 
10%  being  deducted  from  the  premium,  which  is  to  be 
covered  by  the  policy  ?    /  ^ft  * 

7.  A  person  insured  his  house  for  j-  of  its  value  at 
40  cents  per  $100,  paying  a  premium  of  $73.50.  What 
was  the  value  of  the  house  ? 

8.  A  dealer  shipped  a  cargo  of  lumber  from  Portland 
to  New  York ;  the  amount  of  insurance  on  the  lumber, 
at  If  %,  with  the  premium  paid  was  $25200.  What  was 
the  value  of  the  lumber  ? 

9.  A  merchant  had  500  bbl.  of  flour  insured  for  80$  of 
their  cost,  at  3£$,  paying  $107.25  premium.  At  what 
price  per  barrel  must  he  sell  the  flour  to  gain  20$. 

LIFE    INSURANCE. 

667.  Life  Insurance  is  a  contract  by  which  a 
company  agrees  to  pay  a  certain  sum,  in  case  of  the  death 
of  the  insured  during  the  continuance  of  the  policy. 

668.  A  Term  Life  Policy  is  an  assurance  for  one 
or  more  years  specified. 

669.  A  Whole  Lif&  Policy  continues  during  the 
life  of  the  insured. 


INSURANCE.  345 

Premiums  may  be  paid  annually  for  life,  or  in  5,  10,  or  more 
installments  (called  5-payment,  10-payment  policies,  etc.),  or  the 
entire  premium  may  be  paid  in  one  sum  in  advance. 

The  premium  is  computed  at  a  certain  sum  or  rate  per  $1000 
insured,  the  rate  varying  with  the  age  of  the  insured  at  the  time 
the  policy  is  issued. 

A  policy  of  endowment  is  not  in  all  respects  an  insurance  policy, 
but  is  rather  a  covenant  to  pay  a  stipulated  sum  at  the  end  of  a 
certain  period  to  the  person  named,  if  living. 

Most  companies  issue  a  form  of  policy  that  combines  the  princi- 
ples of  Term  Life  Assurance  and  Simple  Endowment,  called  for 
brevity  Endowment  Policy.     Hence, 

670.  An  Endowment  Policy  is  one  in  which 
the  assurance  is  payable  to  the  person  insured  at  the  end 
of  a  certain  number  of  years  named,  or  to  his  heirs  if 
he  die  sooner. 

An  endowment  policy  is  really  two  policies  in  one,  and  the 
assured  pays  the  premiums  of  both. 

671.  A  Dividend  is  a  share  of  the  premiums  or 
profits  returned  to  a  policy-holder  in  a  mutual  life  in- 
surance company. 

672.  A  Table  of  Mortality  shows  how  many  per- 
sons per  1000  at  each  age  are  expected  to  die  per  annum. 

673.  A  Table  of  Hates  shows  the  premium  to  be 
charged  for  $1000  assurance  at  the  different  ages. 

Such  a  table  is  based  upon  the  table  of  mortality,  and  the  proba- 
ble rates  of  interest  for  money  invested,  with  a  margin  or  loading 
for  expenses. 

674.  The  following  condensed  table  gives  data  from 
the  American  Experience  Table  of  mortality,  and  the 
annual  premium  on  the  kinds  of  policies  most  in  use. 


34G 


PEECEXTAGE 


American  Experience  Table— Mortality  and  Premiums. 


i 

3 

ANNUAL  PREMIUM  PER  $1000. 

Lite  Table. 

6* 

t 

3 

, 

E  1TOW- 

MENT 
(AND 

AGE. 

One 

Whole  Life. 

I 

Tear 
Term 

Tekm 

Payments 

Payment 

Payment 

Single 
Payment. 

Life). 
10 

M 

{Net). 

daring 
life. 

lor  10  yr. 
only. 

lor  5  yr. 
only. 

years. 

25 

8.1 

7.75 

$19.89 

$42.56 

$73.87 

$326.58 

$103.91 

26 

8.1 

7.82 

20.40 

43.37 

75.25 

332.58 

104.03 

27 

8.2 

7.88 

23.93 

44.22 

76.69 

338.83 

104.16 

28 

8.3 

7  95 

21.48 

45.10 

78.18 

345.31 

104.29 

29 

8.3 

8.02 

22  07 

46.02 

79.74 

352.05 

104.43 

30 

8.4 

8.10 

22.70 

46.97 

81.36 

359.05 

104.58 

31 

8.5 

8.18 

23.35 

47.93 

83.05 

366.33 

104.75 

33 

8.6 

8.28 

24.05 

49.02 

84.80 

373.89 

1C4.92 

33 

8.7 

8.33 

24.78 

50.10 

86.62 

381.73 

105.11 

34 

8.8 

8.49 

25.56 

51.22 

88.52 

389.88 

105.31 

35 

8.9 

860 

26.38 

52.40 

90.49 

398.34 

105.53 

40 

9.8 

9.42 

31.30 

"59.09 

101.58 

445.55 

106.90 

45 

11.2 

10.73 

37.97 

67.37 

115.02 

501.69 

109.07 

50 

13.8 

13.25 

47.18 

77.77 

131.21 

567.13 

112.68 

The  actual  net  cost  of  insurance  for  a  single  year  at  each  age 
given  in  the  table,  on  the  mortality  assumed,  is  as  many  dollars  and 
tenths  of  a  dollar  as  there  are  deaths,  but  discounted  for  1  year. 
Thus,  at  age  25,  deaths  8.1  per  1000,  net  cost,  which  is  $8.10,  dis- 
counted at  4}2%  by  the  insurance  law,  $7.75.  If  this  sum,  $7.75, 
is  loaded  for  expenses  at,  say  25  % ,  the  total  premium  for  1  year 
is  $9.69,  if  at  40%,  then  it  would  be  $10,85. 

In  a  Term  Life  Policy  the  premium  may  vary,  increasing  slightly 
each  year  of  the  term,  according  to  the  assumed  increasing  liability 
to  decease,  or  it  may  be  averaged  for  the  term  so  as  to  be  the  same 
each  year.  ^ 

The  greater  number  of  policies  now  issued  by  the  companies  are 
on  the  whole  life  plan,  with  equal  annual  premiums. 


INSURANCE.  347 

WRITTEN     EXERCISES. 

675.  To  find  the  amount  of  premium 
1*.  For  a  life  policy  of  $5000  issued  to  a  person  30 
years  old. 
Operation.— $22.70  x  5  =  $113.50. 

2.  For  a  life  policy  of  $7500,  age  being  45. 

~RuLE.—3fultiply  the  premium  for  $1000  assurance  by 
the  number  of  thousands. 

Formula. — Premium  ==  Rate  per  $1000  x  No.  of  thou- 
sands. 

3.  Find  the  annual  premium  for  an  endowment  policy 
of  $10000,  payable  in  10  years,  age  35. 

4.  What  premium  must  a  man  aged  30  pay  annually 
for  life,  for  a  life  policy  of  $5000  ? 

What  premium  annually  for  10  years  ? 
What  premium  annually  for  5  years  ? 
What  premium  in  a  single  payment  ? 

operation.  Analysis. — Multiply  the  rate 

$22.70x5000=    $113.50     Per  thousand  dollars,  found  in 

$40.97  x  5000  ==    $234.85     *he+1Life  Tf e'  ?pPOSite  T  *°' 

by  tlie  number  of  thousands,  ex- 

$81.36  X  5000  ss    $406.80     pressing  the  hundreds,  tens,  and 
$359.05  X  5000  sb  $1795.25      units  decimally. 

5.  What  annual  premium  will  a  man  aged  35  years  pay 
to  secure  an  endowment  policy  for  $5000,  payable  to  him- 
self in  10  years,  or  to  his  heirs,  if  death  occurs  before  ? 

6.  If  he  dies  at  the  beginning  of  the  ninth  year,  how  much 
will  the  assurance  cost,  reckoning  annual  interest  at  6$  ? 

7.  How  much  less  would  he  have  paid  in  the  whole  life 
(annual  payment)  plan,  interest  included  ? 


348  PERCENTAGE. 

8.  A  man  aged  45  insures  his  life  for  $7500  on  the  sin- 
gle-payment  plan,  and  dies  3  yr.  5  mo.  afterward.  How 
much  less  would  his  insurance  have  cost  him  had  he  in- 
sured on  the  annual  payment  plan,  reckoning  int.  at  6#.r 

9.  A  person  aged  27  takes  out  a  10-year  endowment 
policy  for  $5000  ;  the  dividends  reduce  his  annual  pre- 
miums 15%  on  the  average.  Computing  annual  interest  at 
H%  on  his  premiums,  does  he  gain  or  lose,  and  how  much  ? 

10.  A  man  aged  35  years  took  out  a  life  policy  for 
$12000,  on  the  5-payment  plan,  and  died  3  yr.  6  mo. 
afterward.  What  was  gained  to  his  estate  by  insuring, 
computing  compound  interest  on  his  payments  at  7%, 
also  adding  two  dividends  of  $95  each  ? 

TAXES. 

676.  A  Tax  is  a  sum  of  money  assessed  on  the  pe^ 
son,  property,  or  income  of  an  individual,  for  any  public 
purpose. 

677.  A  Poll  Tax  or  Capitation  Tax  is  a  cer- 
tain sum  assessed  on  every  male  citizen  liable  to  taxation. 
Each  person  so  taxed  is  called  a  poll. 

678.  A  Property  Tax  is  a  tax  assessed  on  prop- 
erty, according  to  its  estimated,  or  assessed,  value. 

Property  is  of  two  kinds  :  Real  Property,  or  Real  Es- 
tate, and  Personal  Property. 

679.  Heal  Estate  is  fixed  property  ;  such  as  houses 
and  lands. 

680.  Personal  Propen%ty  is  of  a  movable  nature ; 
6uch  as  furniture,  merchandise,  ships,  cash,  notes,  mort- 
gages, stock,  etc. 


TAXES.  349 

681.  An  Assessor  is  an  officer  appointed  to  deter- 
mine the  taxable  value  of  property,  prepare  the  assess- 
ment rolls,  and  apportion  the  taxes. 

682.  A  Collector  is  an  officer  appointed  to  receive 
the  taxes. 

683.  An  Assessment  Roll  is  a  schedule,  or  list, 
containing  the  names  of  all  the  persons  liable  to  taxation 
in  the  district  or  company  to  be  assessed,  and  the  valua- 
tion of  each  person's  taxable  property. 

684.  The  Rate  of  Property  Tax  is  the  rate  per 
cent,  on  the  valuation  of  the  property  of  a  city,  town, 
or  district,  required  to  raise  a  specific  tax. 

WRITTEN    EXERCISES. 

685.  1.  What  sum  must  be  assessed  to  raise  $836000 
net,  after  deducting  the  cost  of  collection  at  5%  ? 

Operation.— $836000  -f-  .95  =  $880000.    (519.) 

Formula. — Sum  to  be  raised  ■+-  (1  —  Bate  of  Collection) 
=  Sum  to  be  Assessed. 

2.  What  sum  must  be  assessed  to  raise  a  net  amount 
of  811123,  and  pay  the  cost  of  collecting  at  2%  ? 

3.  In  a  certain  district,  a  school-house  is  to  be  built  at 
a  cost  of  $1 8500.  What  amount  must  be  assessed  to  cover 
this  and  the  collector's  fees  at  3%  ? 

4.  The  expense  of  building  a  public  bridge  was  $1260.52, 
which  was  defrayed  by  a  tax  upon  the  property  of  the 
town.  The  rate  of  taxation  was  3}  mills  on  a  dollar, 
and  the  collector's  commission  was  3\%.  What  was  the 
valuation  of  the  property  ? 


350 


PERCENTAGE. 


5.  In  a  certain  town  a  tax  of  $5000  is  to  be  assessed. 
There  are  500  polls,  each  assessed  75  cents,  and  the 
valuation  of  the  taxable  property  is  8370000.  What  will 
be  the  rate  of  property  tax,  and  how  much  will  be  A's  tax, 
whose  property  is  valued  at  $7500,  and  who  pays  for  2  polls  ? 

Operation.— $.75  x  500  =  $375,  amt.  on  polls. 

$5000 -$375=         "      "property. 
$4625  -=-  $370000  =  .0125,  rate  of  taxation. 
$7500  x  .0125  =  $93.75,  A's  property  tax. 
$93.75  +  $1.50  =  $95.25,  A's  whole  tax. 

Eule. — I.  Find  the  amount  of  poll  tax,  if  any,  and 
subtract  it  from  the  whole  amount  to  be  assessed ;  the 
remainder  is  the  property  tax, 

II.  Divide  the  property  tax  by  the  whole  amount  of 
taxable  property  ;  the  quotient  is  the  rate  of  taxation. 

III.  Multiply  each  man's  taxable  property  by  the  rate 
of  taxation,  and  to  the  product  add  his  poll  tax,  if  any  ; 
the  result  is  the  whole  amount  of  his  tax. 

A  table  such  as  the  following  is  a  great  aid  in  calculating  the 
amount  of  each  person's  tax,  according  to  the  ascertained  rate. 


Assessor's  Table.     {Rate  .0087.) 


Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

$1 

$.0087 

$  9 

$.0783 

$  80 

$  .696  , 

$  700 

$  6.09 

2 

.0174 

10 

.087 

90 

.783 

800 

6.96 

3 

.0261 

20 

.174 

100 

.87 

900 

7.83 

4 

.0348 

30 

.261 

200 

1.74  i 

1000 

8.70 

5 

.0435 

40 

.348 

300 

2.61 

2000 

17.40 

6 

.0522 

50 

.435 

400 

3.48 

3000 

26.10 

7 

.0609 

60 

.522 

500 

4.35 

4000 

34.80 

8 

.0696 

70 

.609 

600 

5.22  ; 

5000 

43.50 

TAXES.  351 

6.  Find  by  the  table  the  tax  of  a  person  whose  property 
is  valued  at  $3475,  the  rate  being  .0087. 

Operation.— Tax  on  $3000  =  $26.10 
"  "  400  =  3.48 
"     "         70  =        .609 

"     "      __5= .0435 

"     "  $3475  =  $30.2325,  or  $30.23. 

Find  by  the  table  the  tax  of  a  person  whose  property 

7.  Is  $2596,  and  who  pays  for  5  polls  at  $.50. 

8.  Is  $9785,  polls  3  at  $.75. 

9.  Is  $12356,  polls  4  at  $1.25.       /'  < 

10.  Is  $25489,  polls  5  at  $.95. «  /-  fy  «.  /  /  Ji  ~  *  ? . 
"ll,  A  tax  of  $11384,  •be(siiaes'cost  of  collection  at  3j$/^/J 
is  to  be  raised  in  a  certain  town.  There  are  760  polls 
assessed  at  $1.25  each,  and  the  personal  property  is 
valued  at  $124000,  and  the  real  estate  at  $350000.  Find 
the  tax  rate,  make  an  assessor's  table  for  that  rate,  and 
find  a  person's  tax,  whose  real  estate  is  valued  at  $6750, 
personal  property  at  $2500,  and  who  pays  for  3  polls. 

12.  In  the  above  town,  how  much  is  B's  tax  on  $15000 
real  estate,  $2750  personal  property,  and  5  polls  ? 

13.  What  is  C's  tax  on  $9786  and  1  poll  ? 

14.  How  much  tax  will  a  person  pay  whose  property  is 
assessed  att$7500,  if  he  pays  If  %  village  tax,  \%  State  tax, 
and  \\  mills  on  a  dollar  school  tax  ? 

15.  The  expense  of  constructing  a  bridge  was  $916.65, 
which  was  defrayed  by  a  tax  upon  the  property  of  the 
town.  The  rate  of  taxation  was  2£  mills  on  a  dollar, 
and  the  commission  for  collecting  3% ;  what  was  the 
assessed  valuation  of  the  property  of  the  town  ? 

Note. — Amt.  to  be  raised  4-  by  rate  =  valuation. 


352 


PERCENTAGE. 


6S6.         SYNOPSIS    FOE    REVIEW. 

1.  Corporation.  2.  Charter.  3.  Capital  Stock. 
4.  Certificate  of  Stock,  or  Scrip.  5.  Share. 
6.  Stocks.  7.  Stockholders.  8.  Par  Fa^e. 
9.  Market  Value.    10.  Premium,  Discount, 

1.  Defs.  ^  Brokerage.  11.  #foc&  Broker.  12.  /Sfoc&- 
jobbing.  13.  Installment.  14.  Assessment. 
IT}.  Dividend.  \($.  Net  Earnings.  17.  Bond. 
18.  -Di/l  JST«s  o/  £T.  &  J9<wa>.  19.  Cow- 
_pow.    20.  Currency. 

2.  648.  1  f  tad. 

3.  649.  Jfo.  <?/  £ftam. 

4.  650.  Amt.  of  Investment. 

5.  651,  \  To  find  <(  ifate  %  Income. 

6.  652.  jPnce  to  pay  Income. 

7.  653.  Value  of  Gold  in  Cur. 

8.  654.  J  I  FaJwe  of  Cur.  in  Gold. 


For- 
mula. 


(  1.  Insurance.     2.  Insurer  or    Underwriter. 

1.  Defs.  )       3.  Policy.   4.  Premium.   5.  i^Yre  Insurance. 

I      6.  Marine  or  Inland  Insurance. 

2.  Corresponding  Terms  in  Percentage. 

3.  664.  )  (  Premium. 
S5.  \ 

5.  666.  ) 


.Rate  0/  Insurance. 
Amt.  of  Insurance. 


Formula. 


1.  Defs. 


2.  675. 


1.  Life  Insurance.     2.  Term  Life  Policy.     3. 

TOofo  Z?/<3  Policy.    4.  Endowment  Policy. 

5.  Dividend.  6.  Table  of  Mortality.  7.  TaW<? 

0/  .Rates. 
Rule.     Formula. 


1.  Defs. 


2.  685 

3.  686 


:t 


1.  Ttae.  2.  Poll  Tax.  3.  Property  Tax.  4. 
iteai  Estate.  5.  Personal  Property.  6. 
Assessor.  7.  Collector.  8.  Assessment  Roll. 
9.  .Rate  o/  Property  Tax. 

T    fi  d   ^  ^wm  *0  ^  raised.     Formula. 
|  .4w*.  o/  Taz.     Rule,  I.  II,  III. 


tj^lvl) 


^^p^ 


687.  Exchange  is  the  giving  or  receiving  of  any 
sum  in  one  currency  for  its  value  in  another. 

By  means  of  exchange,  payments  are  made  to  persons  at  a  dis- 
tance by  written  orders,  called  Bills  of  Exchange. 

688.  Exchange  is  of  two  kinds,  Domestic,  or  In- 
land, and  Foreign: 

689.  Domestic  or  Inland  Exchange  relates 
to  remittances  made  between  different  places  in  the  same 
country. 

690.  Foreign  Exchange  relates  to  remittances 
made  between  different  countries. 

691.  A  Bill  of  Exchange  is  a  written  request,  or 
order,  upon  one  person  to  pay  a  certain  sum  to  another 
person,  or  to  his  order,  at  a  specified  time.  An  inland 
bill  of  exchange  is  usually  called  a  Draft. 

692.  A  Set  of  Exchange  is  a  bill  drawn  in  dupli- 
cate or  triplicate,  each  copy  being  valid,  until  the  amount 
of  the  bill  is  paid.  These  copies  are  sent  by  different 
conveyances,  to  provide  against  miscarriage. 

693.  A  Sight  Draft  or  Bill  is  one  which  requires 
payment  to  be  made  "  at  sight,"  that  is,  at  the  time  it  is 
presented  to  the  person  who  is  to  pay  it. 


354  ,  PERCENTAGE. 

694.  A  Time  Draft  or  Bill  is  one  that  requires 
payment  to  be  made  at  a  certain  specified  time  after  date, 
or  after  sight. 

695.  The  Bayer  or  Remitter,  of  a  bill  is  the 
person  who  purchases  it.  The  buyer  and  payee  may  be 
the  same  person. 

696.  The  Acceptance  of  a  bill  or  draft  is  the  agree- 
ment by  the  drawee  to  pay  it  at  maturity.  The  drawee 
thus  becomes  the  acceptor,  and  the  bill  or  draft,  an 
acceptance. 

1.  The  drawee  accepts  by  writing  the  word  "  accepted "  across 
the  face  of  the  bill,  and  signing  it. 

2.  Three  days  of  grace  are  usually  allowed  on  bills  of  exchange, 
as  well  as  on  notes.  When  a  bill  is  protested  for  non-acceptance, 
the  drawer  is  bound  to  pay  it  immediately. 

697.  The  Par  of  Exchange  is  the  estimated  value 
of  the  coins  of  one  country  as  compared  with  those  of 
another.     It  is  either  intrinsic  or  commercial. 

1.  The  Intrinsic  Par  of  Exchange  is  the  comparative  value  of  the 
coins  of  different  countries,  according  to  their  weight  and  purity. 

2.  The  Commercial  Par  of  Exchange  is  the  comparative  value  of 
the  coins  of  different  countries,  according  to  their  market  price. 

698.  The  Course  or  Bate  of  Exchange  is  the 

current  price  paid  in  one  place  for  bills  of  exchange  on 
another  place. 

This  price  varies  according  to  the  relative  conditions  of  trade  and 
commercial  credit  at  the  two  places  between  which  the  exchange  is 
made.  Thus,  if  New  York  is  largely  indebted  to  London,  bills  of 
exchange  on  London  will  bear  a  high  price  in  New  York. 


EXCHANGE.  355 


699.         FORMS  OF  DRAFTS  AND  BILLS. 

A   SIGHT  DEAET. 

$500.  New  York,  July  1, 1874. 

At  sight,  pay  to  the  order  of  William  Thompson,  five 
hundred  dollars,  value  received,  and  charge  to  the  acct.  of 

Henry  J.  Carpenter. 
To  Harris,  Jones  &  Co., 

Cincinnati,  0. 

Other  drafts  have  the  same  form  as  the  above,  except  that  In- 
stead of  the  words  "at  sight,"  " days  after  sight,"  or  " 

days  after  date,"  are  used.    When  the  time  is  after  sight,  it  means 
after  acceptance. 

SET  OF  EXCHANGE. 

£700.  New  York,  August  1, 1874. 

At  sight  of  this  First  of  Exchange  (Second  and  Third 
of  the  same  tenor  and  date  unpaid),  pay  to  the  order  of 
Samuel  Monmouth,  Seven  Hundred  Pounds  Sterling,  for 
value  received,  and  charge  the  same  to  the  account  of 

Morton,  Bliss  &  Co. 

Morton,  Rose  &  Co.,  London. 

The  above  is  the  form  of  the  first  bill ;  the  second  requires  only 
the  change  of  "First"  into  "Second,"  and  instead  of  "Second 
and  Third  of  the  same  tenor,"  etc.,  "  First  and  Third."  The  Third 
Bill  varies  similarly. 

DOMESTIC  OR  INLAND  EXCHANGE. 

The  course  of  exchange  for  inland  bills,  or  drafts,  is  always  ex- 
pressed by  the  rate  of  premium  or  discount.  Time  drafts,  however, 
are  subject  to  bank  discount,  like  promissory  notes,  for  the  term 
of  credit  given.  Hence,  their  cost  is  affected  by  both  the  course  of 
exchange  and  the  rate  of  discount  for  the  time 


356  PERCENTAGE. 

WRITTEN    EXERCISES. 

WO.  What  is  the  cost 

1.  Of  a  sight  draft  on  New  Orleans  for  $1750,  at  \\% 
premium  ? 

Operation.— $1750  x  L01J  =  $1771.87-|,    (512.) 

^  „    ,       „  i  1  +  Rate  of  Premium. 

Formula. — Cost  =  Face  x     ,       0  .     -^. 

(  1  —  Rate  of  Discount. 

2.  Of  a  sight  draft  on  Troy  for  $1590,  at  \\%  discount  ? 

3.  Of  a  draft  on  Boston  for  $1650,  payable  in  60  days 
after  sight,  exchange  being  at  a  premium  of  \\%  ? 

Operation.— $1.0175  =  Course  of  Exchange. 

$.0105  =  Bank  Dis.  on  $1,  for  63  da. 
$1,007    s=  Cost  of  Exchange,  for  $1. 
$1,007  x  1650  =  $1661.55,  value  of  Draft. 

4.  Of  a  draft  on  New  York  at  30  da.  for  $4720,  at  \\% 
premium?    ■ 

5.  Of  a  draft  on  New  Orleans,  at  90  da.,  for  $5275,  int. 
being  7%,  and  exchange  \%  discount  ? 

6.  Find  the  cost  in  Philadelphia  of  a  draft  on  Denver, 
at  90  da.,  for  $6400,  the  course  of  exchange  being  lOlf  ? 

7.  What  must  be  paid  in  New  York  for  a  draft  on 
San  Francisco,  at  90  da.,  for  $5600,  the  course  of  ex- 
change being  102£%  ?    4%  J 2  .  X^> 

"701.  Find  the  Face 

1.  Of  a  draft  on  St.  Louis,  at  90  da.,  purchased  for 
$4500,  exchange  being  at  101^. 

Operation.— $1,015    =  Course  of  Exchange. 

$.0155  =  Bank  Dis.  of  $1,  for  93  da.,  at  6%. 
$.9995  =  Cost  of  Exchange  of  $1. 
$4500  -r-  .9995  =  $4502.25.     (520.) 


EXCHANGE.  357 

2.  Of  a  draft  on  Eichmond  at  60  da.  sight,  purchased 
for  $797.50,  interest  7$,  premium  %\%. 
^3.  Of  a  sight  draft  bought  for  $711.90,  discount  1\%.  J7fr* 

4.  A  commission  merchant  sold  2780  lb.  of  cotton  at  "i. 
11 J  cents  a  pound.  If  his  commission  is  %\%,  and  the /t  0*2- 
course  of  exchange  98£%,  how  large  a  draft  can  he  buy  to  fffri 
remit  to  his  consignor  ?  < /f^Tj'k 

5.  The  Broadway  Bank  of  New  York  haying  declared 
a  dividend  of  5%,  a  stockholder  in  Chicago  drew  on  the 
bank  for  the  sum  due  him,  and  sold  the  draft  at  a  pre- 
mium of  \\%,  thus  realizing  $2283.18f  from  his  dividend. 
How  many  shares  did  he  own  ?  '.p  7   • 

6.  A  man  in  Rochester  purchased  a  draft  on  Louisville*,  ^ 
Ky.,  for  $5320,  drawn  at  60  days,  paying  $5141.78.    What 

was  the  course  of  exchange  ? 

7.  Received  from  Savannah  250  bales  of  cotton,  each,^7~< 
weighing  520  pounds,  and  invoiced  at  12£  cents  a  pound. 
Sold  it  at  an  advance  of  25$,  commission   1\%,  and 
remitted  the  proceeds  by  draft.     What  was  the  face  of 

the  draft,  exchange  being  \%  discount  ?    /  b  2_  i """  A  /, : 

FOREIGN   EXCHANGE. 

702.  Money  of  Account  consists  of  the  denomi- 
nations or  divisions  of  money  of  any  particular  country, 
in  which  accounts  are  kept. 

The  Act  of  March  3,  1873,  provides  that  "the  value  of  foreign 
coin,  as  expressed  in  the  money  of  account  of  the  United  States, 
shall  be  that  of  the  pure  metal  of  such  coin  of  standard  value  ;  and 
the  values  of  the  standard  coins  in  circulation,  of  the  various  na- 
tions of  the  world,  shall  be  estimated  annually  by  the  Director  of 
the  Mint,  and  be  proclaimed  on  the  first  day  of  January  by  the 
Secretary  of  the  Treasury." 


358 


PEECENTAGE. 


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EXCHANGE.  359 

704.  Sterling  Bills  or  Sterling  Exchange 

are  bills  on  England,  Ireland,  or  Scotland.  Such  bills 
are  negotiated  at  a  rate  fixed  without  reference  to  the  par 
of  exchange. 

Formerly  such  bills  were  quoted  at  a  certain  rate  fo  above  the 
old  par  value  of  a  pound  sterling,  which  was  $4.44|.  As  this  was 
entirely  a  fictitious  value,  and  always  about  9%  below  the  real 
value,  the  course  of  exchange  always  appeared  to  be  heavily  against 
this  country,  and  thus  tended  to  impair  its  credit.  By  the  Act  of 
March,  1873,  "all  contracts  made  after  the  first  day  of  January, 
1874,  based  on  an  assumed  par  of  exchange  with  Great  Britain  of 
fifty-four  pence  to  the  dollar,  or  $4.44;*  to  the  sovereign  or  pound 
sterling,"  are  declared  null  and  void.  The  par  of  exchange  between 
Great  Britain  and  the  United  States  is  fixed  at  $4.80(35. 

705.  Exchanges  ivith  Europe  are  effected 
chiefly  through  the  following  prominent  financial  circles: 
London,  Paris,  Antwerp,  Amsterdam,  Hamburg,  Frank- 
fort, Bremen,  and  Berlin. 

In  exchange  on  Paris,  Antwerp,  and  Switzerland,  the  unit  is 
the  franc,  and  the  quotation  shows  the  number  of  francs  and 
centimes  to  the  dollar,  Federal  Money.  In  exchange  on  Amster- 
dam, the  unit  is  the  guilder,  quoted  at  its  value  in  cents  ;  on  Ham- 
burg, Frankfort,  Bremen,  and  Berlin,  the  quotation  shows  the  value 
of  four  reichsmarks  (marks)  in  cents. 

WRITTEN     EXAMPLES. 

706.  Find  the  cost 

1.  Of  a  bill  of  exchange  on  London  at  3  days'  sight, 
for  £393  15s.  6d,  exchange  being  quoted  at  4.89£,  and 
gold  at  1.10J. 

OPERATION. 

£393  15s.  6d.  =  £393.775. 

$4,895  x  393.775  =  $1927.529,  gold  value  of  bill.  * 

$1927.529  x  1.10 J-  ==  $2122.69,  value  in  currency. 


360  PERCENTAGE. 

2.  Of  a  bill  of  exchange  on  Liverpool,  for  £473  5s.  9d 
par  value,  in  gold. 

3.  Of  a  bill  of  £625  4s.  3d.  sterling,  at  4.83 J,  gold  1.09J . 

4.  Of  a  bill  on  Paris  for  495  francs,  at  5.15  francs  to 
the  dollar,  in  gold. 

Operation.— 495  -r-  5.15  =  $96.12,  gold  value  of  the  bill. 

Analysis. — Since  5.15  francs  cost  $1,  495  francs  will  cost  as  many 
dollars  as  5.15  francs  are  contained  times  in  495  francs,  or  $96.12. 

5.  Of  a  bill  on  Antwerp  for  697|  francs,  at  5.17J  francs 
to  the  dollar,  in  gold. 

6.  Of  a  bill  on  Geneva,  Switzerland,  for  1655  francs, 
at  5.15£,  in  currency,  gold  being  1.0  9-J. 

7.  Of  a  bill  on  Frankfort  for  650  marks,  at  94$,  in  gold. 

Operation.— $.94375  -*-  4  x  650  =  $153.36. 

Analysis. — Since  $.94f  is  the  value  of  4  marks,  650  marks  are 
worth  650  times  \  of  $.94f,  or  $153.36. 

s-    8.  Of  a  bill  on  Berlin  for  1750  marks,  quoted  at  96£, 
in  gold. 

9.  Of  a  bill  of  Hamburg  for  2155  marks,  at  95f,  in 
currency,  gold  being  1.1 0£. 

10.  Of  a  bill  on  Amsterdam  for  2500  guilders,  quoted 

at  41|,  brokerage  \%. 

Operation.— $.41625  x  2500  =  $1040.625. 

$1040.625  x  .00£  =  $2.60,  brokerage. 
$1040.625  4- $2.60  =  $1043.225,  cost  of  bill. 

11.  Of  a  bill  on  Amsterdam  for  1950  guilders,  at  41J. 

12.  Bought  exchange  on  Amsterdam,  at  41£,  for  3750 
guilders  ;  on  Hamburg,  at  95f  for  1000  marks  ;  and  on 
London  for  £500,  at  $4.85.  What  was  the  cost  of  the 
whole  in  currency,  gold  selling  at  109 J? 


EXCHANGE.  361 

13.  What  will  it  cost  to  remit  directly  from  Boston  to 
Amsterdam,  12560  guilders,  at  41f? 
H    14.   What  will  be  the  cost  of  remitting  13550  marks 
from  New  York  to  Frankfort,  exchange  selling  at  94J,-  $* 
and  gold  at  109 J  ;  brokerage,  \%  ?  /  j  yi  0) 

707.  What  will  be  the  face  /  09tr 

1.  Of  a  bill  of  exchange  on  London  that  can  be  bought  3>  * 
for  $5500,  in  currency,  exchange  selling  at  4.86,  and  gold 

at  1.10  ? 
Operation.— $5500  currency  h-  1.10  =  $5000,  gold.    (519.) 
$5000  -r-  $4.86  =  1028.806  + . 
£1023.806  =  £1028  16s.  1M. 

2.  Of  a  bill  on  Manchester,  England,  that  can   be 
bought  for  $7500,  gold  ;  rate  of  exchange,  4.86  ? 

3.  Of  a  bill  on  Berlin  that  cost  $4000  in  gold,  ex- 
change 93}? 

Operation.— ($4000  -*-  $.9375)  x  4  =  17066f  marks. 
Analysis.— Since  $.93|  will  buy  4  marks,  $4000  will  buy  4  times 
as  many  marks  as  $.93  J  is  contained  times  in  $4000,  or  17066|  marks. 

4.  Of  a  bill  on  Hamburg  that  cost  $550  in  gold,  ex- 
change 94-J? 

5.  Of  a  bill  on  Frankfort  that  cost  $395.75  in  gold, 
exchange  95£? 

6.  Of  a  bill  on  Geneva,  Switzerland,  that  cost  $325  in 

gold,  exchange  at  5.17? 

Operation.— 5.17  fr.  x  325  =  1680.25  francs. 
Analysis.  -If  $1  will  buy  5.17  francs,  $325  will  buy  325  times 
5.17  francs,  or  1680.25  francs. 

7.  A  merchant  in  New  Orleans  gave  $6186,  currency, 
for  a  bill  on  Paris,  at  5.15f     What  was  its  face  ? 

8.  What  is  the  face  of  a  bill  on  Antwerp,  that  may  be 
purchased  in  New  York  for  $2500,  exchange  at  5.16J? 

16 


362  /^^EECENTAGE. 

.TION    OF    EXCHANGE. 


RBITRATK 


708.  Arbitration  of  Exchange  is  the  process  of 
computing  the  cost  of  exchange  between  two  places  by 
means'  of  one  or  more  intermediate  exchanges.  Such  ex- 
change is  said  to  be  indirect,  or  circuitous. 

By  this  computation  the  relative  cost  of  direct  and  indirect  ex- 
change is  ascertained.  Sometimes,  owing  to  the  course  of  exchange 
between  different  places,  it  is  more  advantageous  to  remit  by  the 
latter  than  by  the  former. 

Arbitration  is  either  simple  or  compound. 

709.  Simple  Arbitration  is  that  in  which  there 
is  but  one  intermediate  place. 

710.  Compound  Arbitration  is  that  in  which 
there  are  several  intermediate  places. 

WRITTEN      EXERCISES. 

711.  1.  I  owe  1500  marks  to  a  merchant  in  Frankfort. 
Should  I  remit  directly  from  New  York,  or  through  Lon- 
don, exchange  on  Frankfort  being  94,  on  London  4.87J, 
and  in  the  latter  place  on  Frankfort  20.75  marks  to  the 
pound,  and  the  London  brokerage  \%  ? 

Operation. — $.94  x  1500-5-4= $352.50,  cost  of  direct  exchange. 
1500  marks  -j-  20.75  marks  =  £72.29. 
£72.29  +  \%  =£72.38. 
$4,874  x  72.38  =  $352.85. 
$352.85  —  $352.50  =  $.35,  loss  by  ind.  exchange. 

2.  What  will  it  cost  to  remit  from  Boston  to  Berlin 
750  marks,  by  indirect  exchange,  through  Paris,  exchange 
in  New  York  on  Paris  being  at  5.15,  and  4  marks 
at  Paris  being  worth  4.91  francs,  the  brokerage  being 


EXCHANGE.  363 

3.  "What  will  it  cost  to  remit  2500  guilders  from  New- 
York  to  Amsterdam,  through  London  and  Paris,  the  rates 
of  exchange  being  as  follows  :  at  New  York  on  London 
4.83,  at  London  on  Paris  24.75  francs  to  the  pound,  and 
at  Paris  on  Amsterdam  2.09  francs  to  the  guilder,  broker- 
age  at  London  and  Paris  \%  each  ? 

OPERATION. 

%  x  —  2500  guilders. 

1  guilder       =2.09  francs. 

1  franc  (net)  =  1.00  J-  (with  brokerage). 

24.75  francs  =s  £1. 

£1  (net)  =  £1.00£  (with  brokerage). 

£1  =  $4.83. 

2500  x  2.09  x  l.OOi  x  1.00?  x  4.83 
Hence, ^ 1 ,    or 

By  cancellation,      10°*  10x1.00^x1.00^  1.61  =  ^^ 

Analysis. — Since  the  members  of  each  equation  are  equal,  the 
product  of  the  corresponding  members  of  any  number  of  equations 
are  equal ;  hence,  the  product  of  all  the  second  members  divided  by 
the  product  of  all  the  first  members  except  one,  must  give  that 
member,  which  is  the  value  required. 

4.  A  merchant  in  St.  Louis  directs  his  agent  in  New 
York  to  draw  upon  Philadelphia  at  1%  discount,  for 
$1500  due  from  the  sale  of  mdse.  ;  he  then  draws  upon 
the  New  York  agent,  at  2%  premium,  for  the  proceeds, 
after  allowing  the  agent  to  reserve  \%  commission.  What 
sum  does  he  realize  from  his  mdse.  ? 

OPETtATION. 

(x)  St.  L.  =  1500  Philadelphia. 
100  Phil.   =      99  N.York. 
100  N.  Y.  =    102  St.  Louis. 
1  =  .995  (net  proceeds). 

By  cancellation,    .  15  x  99  x  102  x  .995 =$1507. 13. 


364  PERCENTAGE. 

Analysis .— $100  on  Philadelphia  =  $99  on  N.  T.,  and  $100  on 
N.  Y.  =  $102  on  St.  Louis  ;  and  since  the  agent  reserves  \%  com- 
mission, $1  realized  =  $.995  net  proceeds.  Arranging,  canceling, 
and  multiplying,  we  find  the  result  to  be  $1507.13. 

Eule. — I.  Represent  the  required  sum  by  (x),  luith  the 
proper  unit  of  currency  affixed,  and  place  it  equal  to  the 
given  sum  on  the  right. 

II.  Arrange  the  given  rates  of  exchange  so  that  in  any 
tivo  consecutive  equations  the  same  unit  of  currency  shall 
stand  on  opposite  sides. 

III.  When  there  is  commission  for  draiving,  place  1  minus 
the  rate  on  the  left  if  the  cost  of  exchange  is  required,  and 
on  the  right  if  proceeds  are  required ;  and  when  there  is 
commission  for  remitting,  place  1  plus  the  rate  on  the 
right,  if  cost  is  required,  and  on  the  left,  if  proceeds  are 
required. 

IV.  Divide  the  product  of  the  numbers  on  the  right  by 
the  product  of  the  numbers  on  the  left,  canceling  equal  fac- 
tors, and  the  result  will  be  the  required  sum. 

Commission  for  drawing  is  commission  on  the  sale  of  a  draft ; 
commission  for  remitting  is  commission  on  the  purchase  price  of  a 
draft. 

The  above  method  of  operation  is  sometimes  called  the  Chain  Rule. 

5.  If  at  New  York  exchange  on  London  is  4.84£,  and 
at  London  on  Paris  it  is  25.73  francs  to  the  £,  what  is 
the  arbitrated  course  of  exchange  between  New  York  and 
Paris? 

6.  If  in  London  exchange  on  Paris  is  25.71,  and  in 
New  York  on  Paris  it  is  5.15J,  what  is  the  arbitrated 
course  of  exchange  between  New  York  and  London  ? 


EXCHANGE.  365 

7.  A  banker  in  New  York  remits  $5000  to  Liverpool 
by  indirect  exchange,  through  Paris,  Hamburg,  and  Am- 
sterdam, the  rates  being  as  follows  :  in  New  York  on 
Paris  5.18  fr.  to  the  dollar,  in  Paris  on  Hamburg  1.22  fr. 
to  the  mark,  in  Hamburg  on  Amsterdam  1.70  mark  to  the 
guilder,  and  in  Amsterdam  11.83  guilders  to  the  pound 
sterling.  How  much  sterling  will  he  have  in  bank  at 
Liverpool,  and  how  much  does  he  gain  by  indirect  ex- 
change, sterling  being  worth  in  New  York  4.83£  ? 

8.  A  merchant  in  Philadelphia  owes  a  correspondent 
in  Paris  35000  francs.  Direct  exchange  on  Paris  is  5.15  ; 
but  exchange  on  London  is  4.83,  and  London  exchange 
on  Paris  is  25. 12.  Allowing  \%  commission  for  brokerage 
at  London,  which  is  the  more  advantageous  way  to  remit, 
and  by  how  much  ? 

9.  An  American  resident  at  Amsterdam  wishing  to 
obtain  funds  from  the  U.  S.  to  the  amount  of  $4500, 
directs  his  agent  in  London  to  draw  on  Philadelphia,  and 
remit  the  proceeds  to  him  in  a  draft  on  Amsterdam,  ex- 
change on  London  in  Phil,  selling  at  4.87J,  and  in  Lon- 
don on  Amsterdam  11.17-J-  guilders  to  the  pound  sterling. 
If  the  agent  charges  commission  at  \%  both  for  drawing 
and  remitting,  how  much  better  is  this  arbitration  than 
to  draw  directly  on  the  U.  S.  at  41£  cents  per  guilder  ? 

10.  A  speculator  residing  in  Cincinnati,  having  pur- 
chased 165  shares  of  railroad  stock  in  New  Orleans,  at 
75%,  remits  to  his  agent  in  N.  York  a  draft  purchased  at 
2%  premium,  directing  the  agent  to  remit  the  sum  due  on 
N.  Orleans.  Now,  if  exchange  on  N.  Orleans  is  at  \%  dis- 
count in  N.  Y.,  and  the  agent's  commission  for  remitting 
is  \%,  how  much  does  the  stock  cost  in  Cincinnati  ? 


3G6  PERCENTAGE 


0 


CUSTOM-HOUSE    BUSINESS. 

712.  A  Custom-House  is  an  office  established  by 
government  for  the  transaction  of  business  relating  to  the 
collection  of  customs  or  duties,  and  the  entry  and  clear- 
ance of  vessels. 

713.  A  JPort  of  Entry  is  a  seaport  town  in  which 
a  custom-house  is  established. 

714.  The  Collector  of  the  JPort  is  the  officer  ap- 
pointed by  government  to  attend  to  the  collection  of 
duties  and  to  other  custom-house  business. 

715.  A  Clearance  is  a  certificate  given  by  the  Col- 
lector of  the  port,  that  a  vessel  has  been  entered  and 
cleared  according  to  law. 

By  the  entry  of  a  vessel  is  meant  the  lodgment  of  its  papers  in 
the  custom-house,  on  its  arrival  at  the  port. 

716.  A  Manifest  is  a  detailed  statement,  or  invoice, 
of  a  ship's  cargo. 

No  goods,  wares,  or  merchandise  can  be  brought  into  the  United 
States  by  any  vessel,  unless  the  master  has  on  board  a  full  mani- 
fest, showing  in  detail  tho  several  items  of  the  cargo,  the  place 
where  it  was  shipped,  the  names  of  the  consignees,  etc. 

717.  Duties  or  Customs  are  taxes  levied  on  im- 
ported goods. 

The  general  object  of  such  taxes  is  the  support  of  government, 
but  they  are  also  designed  sometimes  to  protect  the  manufacturing 
industry  of  a  country  against  foreign  competition. 

718.  A  Tariff  is  a  schedule  showing  the  rates  of 
duties  fixed  by  law  on  all  kinds  of  imported  merchandise. 

Duties  are  of  two  kinds,  Specific  and  Ad  Valorem. 


CUSTOM-HOUSE     BUSINESS.  367 

719.  A  Specific  Duty  is  a  fixed  sum  imposed  on 
articles  according  to  their  weight  or  measure,  but  without 
regard  to  their  value. 

■720.  An  Ad  Valorem  Duty  is  an  import  duty 

assessed  by  a  j)ercentage  of  the  value  of  the  goods  in-  the 

country  from  which  they  are  brought. 

Before  computing  specific  duties,  certain  deductions,  or  allow- 
ances, are  made,  called  Tare,  Leakage,  Breakage,  etc. 

721.  Tare  is  an  allowance  for  the  weight  of  the  box,  cask, 
bag,  etc.,  that  contains  the  merchandise. 

722.  Leakage  is  an  allowance  for  waste  of  liquors  imported 
in  casks  or  barrels. 

723.  Breakage  is  an  allowance  for  loss  of  liquors  imported 
in  bottles. 

724.  Gross  Weight  or  Value  is  the  weight  or 
value  of  the  goods  before  any  allowance  is  made. 

725.  Net  Weight  or  Value  is  the  weight  or  value 
of  the  goods  after  all  allowances  have  been  deducted. 

WRITTEN      EXERCISES. 

726.  Find  the  Duty 

1.  On  355  yds.  of  carpeting,  invoiced  at  lis.  6d.  per 
yd.,  the  duty  being  50%. 

Operation.— lis.  6d.  =  £.575. 

£.575  x  355  =  £204.125. 

$4.8665  (par  value  of  £1)  x  204 125  =  $993.37. 

$993.37  x  .50  =  $496.68,  duty.    (510.) 

2.  On  50  hhd.  of  sugar,  each  containing  500  lb.,  at  & 
cts.  per  lb.  ;  duty  If  cts.  per  lb. 

3.  On  350  boxes  of  cigars,  each  containing  100  cigars, 
invoiced  at  $7.50  per  box  ;  weight,  12  lb.  per  1000  ;  duty, 
$2.50  per  lb.,  and  25%  ad  valorem. 

V 


368  PERCENTAGE. 

4.  A  wine  merchant  in  New  York  imported  from  Havre 
100  doz.  quart  bottles  of  champagne,  at  $13  per  doz.,  and 
25  casks  of  sherry  wine,  each  containing  30  gals.,  at  $2.50 
per  gal.  What  is  the  duty,  the  rate  on  the  champagne 
being  $6  per  dozen,  and  on  the  sherry  60  cents  per  gal., 
and  25$  ad  valorem  ? 

5.  Imported  from  Geneva  25  watches  invoiced  at  $125 
each,  and  15  clocks,  at  $37.50.  What  was  the  duty,  the 
rate  being  on  clocks  25$,  and  on  watches,  35$  ad  valorem? 

6.  A  liquor  dealer  receives  an  invoice  of  120  dozen  pint 
bottles  of  porter,  rated  at  $.75  per  dozen.  If  %\%  of  the 
bottles  are  found  broken,  what  will  be  the  duty  at  36  cts. 
per  gallon  ? 

7.  H.  B.  Claflin  &  Co.  imported  20  cases  of  bleached 
muslins,  each  case  containing  175  pieces  of  24  yards 
each,  \\  yards  wide.  What  was  the  duty  at  5-J-  cts.  per 
square  yard  ? 

8.  What  was  the  duty  on  10  cases  of  shawls,  average 
weight  of  each  case  213£  lb.,  invoiced  at  19375  francs ; 
rate  of  duty,  50  cts.  per  lb.  and  35$  ad  valorem  ?    If  I  pay 

«v/  ~oior  the  invoice  with  a  bill  of  exchange  bought  at  5.15£, 
and  pay  charges  amounting  to  $67.50  currency,  what  do 
\']f+*  the  shawls  cost  me  in  currency,  gold  selling  at  1.10  ? 

9.  Olmsted   &   Taylor,   of   New    York,   import    from 
%  •  r ^Switzerland  1  case  of  watches,  invoiced  at  7125  francs ; 

*i  f  duty,  25$;    charges,   13.50    francs;    commissions,   2|$. 

AY  bat  was  the  cost  of  the  watches  in  U.  S.  gold  ? 
f  4>l>^    10.  imported  from  England  5  cases  of  cloths  and  cassi- 
♦  7.'  i  meres,  net   weight,  695  lb.;   value  as  per  invoice,  £375 
10s.     What  was  the   duty  in  American  gold,  the  rate 
being  50  cts.  per  lb.  and  35$  ad  valorem  ?  ^\ 


-;■ 


EQUATION     OF     PATMEJfTS.  369 


EQUATION    OF    PAYMENTS. 

727.  Equation  of  Payments  is  the  process  of 
finding  the  average  time  for  the  payment  of  several  sums 
of  money  due  at  different  times,  without  loss  to  debtor 
or  creditor. 

728.  The  Equated  Time  is  the  date  at  which  the 
several  debts  may  be  discharged  by  one  payment. 

729.  The   Term  of  Credit  is  the  time  at  the 

expiration  of  which  a  debt  becomes  due. 

• 

730.  The  Average  Term  of  Credit  is  the  time 
at  the  end  of  which  the  several  debts  due  at  different 
dates,  may  all  be  paid  at  once,  without  loss  to  debtor  or 
creditor. 

ORAL      EXERCISES. 

731.  1.  The  interest  of  $100  for  3  mo.  equals  the 
interest  of  $50  for  how  many  months  ? 

Analysis. — At  the  same  rate,  the  interest  of  $100  equals  the 
interest  of  $50,  or  one-half  of  $100,  for  twice  the  time,  or  6  mo. 

2.  The  interest  of  $20  for  4  mo.  equals  the  interest  of 
$10  for  how  many  mo.  ?  Equals  the  interest  of  $5  for 
how  many  mo.?     Of  $1  ?     Of  $40  ?     Of  $100  ? 

3.  The  interest  of  $25  for  6  mo.  equals  the  interest  of 
$5  for  how  many  mo.  ?   #Of  $10  ?     Of  $1  ? 

4.  The  interest  of  $10  for  6  mo.,  and  of  $100  for  2  mo., 
taken  together,  equals  the  interest  of  $1  for  how  many 
months  ? 


370  PERCENTAGE. 

5.  If  I  borrow  150  for  3  mo.,  for  how  many  months 

should  I  lend  $100  to  repay  an  equal  amount  of  interest? 

Analysis. — The  interest  of  $50  for  3  mo.  is  the  same  as  the 
interest  of  $1  for  50  times  3  mo.,  or  150  mo. ;  and  the  interest  of  $1 
for  150  mo.  is  the  same  as  the  interest  of  $100  for  jjj  of  150  mo., 
or  1|  mo. 

6.  If  I  lend  $200  for  3  mo.,  for  how  long  a  time  should 
I  have  the  use  of  $50  to  balance  the  favor  ? 

7.  If  A  borrows  of  B  $1000  for  3  mo.,  what  sum 
should  A  lend  B  for  9  mo.  to  discharge  the  obligation  ? 

732.  Peikciple. — The  interest  and  rate  remaining  the 
same,  the  greater  the  principal  the  less  the  time,  and  the 
less  the  principal  the  greater  the  time. 

WRITTEN     EXERCISES. 

733.  Find  the  average  term  of  credit 

1.  Of  1300  due  in  cash,  $500  due  in  3  mo.,  $750  due 
in  8  mo.,  and  $950  due  in  10  mo. 

operation.  Analysis.— On  $300,  the  first 

30  0  X      0  =           0  payment,    there  is    no    interest, 

since  it  is  due  in  cash  ;  the  int. 

500  X     6  —  1500  of  $500  for  3  mo.,  is  the  same  as 

750  X      8  =  6000  the  int.  of  $1  for  1500  mo.;  the 

950  Xl0  =  9500  int.  of  $750  for  8  mo.  is  the  same 


oKnA  M7nnn  as  tliat  of  $1  for  coo°  mo-;  anci 

;WUUU  the  int.  of  $950  for  10  mo.  is  the 

6  \  mo.      same  as  the  int.  of  $1  for  9500  mo. 

Therefore,  the  whole  amt.  of  int. 

is  that  of  $1  for  1500  mo.  +  6000  mo.  +  9500  mo.,  or  17000  mo. ;  but 

the  whole  debt  is  $2500  ;  and  the  int.  of  $1  for  17000  mo.  is  equal 

to  the  int.  of  $2500  for  ^^  of  17000  mo.,  or  6*  mo. 

2.  Find  the  average  term  of  credit  of  $800  due  in  1  mo., 
$750  due  in  4  mo.,  and  $1000  due  in  G  mo. 


EQUATION     OF     PAYMENTS.  371 

Bule. — I.  Multiply  each  payment  by  its  term  of  credit, 
and  divide  the  sum  of  the  products  by  the  sum  of  the  pay- 
ments; the  quotient  is  the  average  term  of  credit. 

II.  (To  find  the  equated  time  of  payment,)  Add  th& 
average  term  of  credit  to  the  date  at  which  the  several 
credits  begin. 

3.  On  the  first  day  of  December,  1876,  a  man  gave 
3  notes,  the  first  for  $500  payable  in  3  mo. ;  the  second 
for  $750  payable  in  6  mo.  ;  and  the  third  for  $1200  paya- 
ble in  9  mo.  What  was  the  average  term  of  credit,  and 
the  equated  time  of  payment  ? 

4.  Bought  merchandise  Jan.  1,  1875,  as  follows  :  $350 
on  2  mo.,  $500  on  3  mo.,  $700  on  6  mo.  What  is  the 
equated  time  of  payment  ? 

5.  A  person  owes  a  debt  cf  $1680  due  in  8  months,  of 
which  he  pays  £  in  3  mo.,  }  in  5  mo.,  £  in  6  mo.,  and 
-J-  in  7  mo.     When  is  the  remainder  due  ? 

6.  Bought  a  bill  of  goods,  amounting  to  $1500  on  6 
months'  credit.  At  the  end  of  2  mo.,  I  paid  $300  on 
account,  and  2  mo.  afterward,  paid  $400  on  account,  at 
the  same  time  giving  my  note  for  the  balance.  For  what 
time  was  the  note  drawn  ? 

operation.  Analysis.  —  $300   paid 

300x4  =  1200  4  mo.  before  it  is  due,  and 


400  x  2  =     800 


$400,   2  mo.  before  it  is 
due,  are  equivalent  to  the 


8  0  0  )2000  use  of  $1  for  2000  months, 

2-1  or  tlie  use  of  $800  (tne 

'  .         .  "  balance)  for  2 1  mo.  be vond 

(omo.-4mo.)  +  2imo.=:4£mo.    the  original  time.    Hence, 

the  note  was    drawn  for 
4|  mo.  after  the  second  payment. 


372  PERCENTAGE. 

7.  On  a  debt  of  $2500  due  in  8  mo.  from  Feb.  1,  the 
following  payments  were  made  :  May  1,  $250,  July  1, 
$300,  and  Sept.  1,  $500.     When  is  the  balance  due  ? 

8.  Find  the  average  term  of  credit,  and  the  equated 
time  of  payment  from  Dec.  15,  of  $225  due  in  35  da., 
$350  due  in  60  da.,  and  $750  due  in  90  da. 

9.  Dec.  1,  1874,  purchased  goods  to  the  amount  of 
$1200,  on  the  following  terms  :  25$  payable  in  cash, 
30$  in  3  mo.,  20$  in  4  mo.,  and  the  balance  in  6  mo. 
Find  the  equated  time  of  payment,  and  the  cash  value  of 
the  goods,  computing  discount  at  7$. 

*734.  To  find  the  equated  time  when  the  terms  of 
credit  begin  at  different  dates. 

1.  J.  Prince  bought  goods  of  W.  Sloan  as  follows : 
June  1, 1874,  amounting  to  $350  on  2  mo.  credit ;  July  15, 
1874,  $400,  on  3  mo.  credit ;  Aug.  10,  $450,  on  4  mo. 
credit ;  Sept.  12,  $600,  on  6  mo.  credit.  What  is  the 
equated  time  of  payment  ? 


$350 

due 

VJTXUX1 

Aug.  1, 

350  x   0  =     0 

400 

" 

Oct.  15, 

400  x  75  =   30000 

450 

** 

Dec.  10, 

450  x  131  =   58950 

600 

«< 

Mar.  12, 

600  x  223  =  133800 
1800     1800)222750 

123f 
Hence  the  equated  time  is  124  da.  from  Aug  1,  or  Dec.  3. 

Analysis.— Computing  the  terms  of  credit  from  Aug.  1,  the 
earliest  date  at  which  any  of  the  debts  become  due,  we  find  the 
terms  of  credit  to  be  from  Aug.  1  to  Oct.  15,  75  da. ;  to  Dec.  10, 
131  da.,  and  to  March  12,  223  da.  The  average  term  of  credit  is 
therefore  124  da.  from  Aug.  1,  and  the  equated  time  Dec.  3. 


EQUATION     OF     PAYMENTS.  373 

Proof. — Assume  as  the  standard  time  the  latest  date,  March  12. 
The  operation  will  then  be  as  follows  : 

350  x  223  =  78050 
400  x  148  =  59200 
450     x       92    =    41400 

600     x         0     = 0 

1800)178650 
99J: 
Hence,  the  equated  time  is  99  da.  previous  to  March  12,  or  Dec.  3, 

2.  Peake  &  Co.  sell  to  Wm.  Jones  the  following  bills 
of  goods  :  March  1,  1875,  on  60  da.,  $800  ;  April  15,  on 
30  da.,  $350 ;  May  20,  on  4  mo.,  $3800. 

What  is  the  equated  time  for  settlement  ? 

Rule. — I.  Find  the  date  at  ivhich  each  debt  becomes  due. 

II.  From  the  earliest  of  these  dates  as  a  standard  com- 
pute the  time  to  each  of  the  others. 

III.  Then  find  the  average  term  of  credit  and  equated 
time  as  in  (733). 

Proof. — Compute  the  terms  of  credit  backward  from  the 

latest  date,  and  subtract  the  average  time  from  that  date 

for  the  equated  time. 

If  the  earliest  date  is  not  the  first  of  the  month,  it  is  more  con- 
venient to  assume  the  first  of  the  month  as  the  standard  date 

3.  Bought  mdse.  as  follows  :  Jan.  15,  1876,  on  4  mo., 
$375  ;  Feb.  3,  on  60  da.,  $550 ;  March  25,  on  4  mo., 
$1100  ;  April  2,  on  30  da.,  $250.     Find  the  equated  time. 

4.  Ira  Blunt,  of  Gadsden,  Ala.,  bought  of  Opdyke  & 
Co.  the  following  bills  of  goods  on  4  months'  credit : 

Jan.  1,  1874,  $650  ;  Feb.  10,  $380  ;  March  12,  $900 ; 
March  18,  $350 ;  April  3,  $600. 

April  5,  he  discounted  his  bills  at  %%  per  month.  Find 
the  equated  time  of  payment,  and  the  discount. 


374  PEIICE^TAGE. 

5.  James  Smith  to  Thomas  Brown,  Dr. 

March  10,  1374        To  mdse.        $835. 


18, 


"      26, 


"      "  320. 

i  a       c4  475t 

(  "       "  600. 

'  "       "  250. 

credit  on  each  of  the  bills,  what  is 


April      5, 
12, 
Allowing  30  days 
the  equated  time  of  payment  ? 

6.  Purchased  goods  as  follows  : 

Sept.  15,  1875,  a  bill  of  $275,       on  3  mos. 
Oct.   10,     "  "  351.50,  "  60  da. 

«     28,     "  "  415.75,  "  30  da. 

Nov.    3,     "  "  500,        "    4  mos. 

Dec.  15,     "  "  710,        "    3  mos. 

What  was  due  on  this  account  Aug.  10,  1876,  com- 
puting interest  at  7%  ? 

7.  I  have  four  notes,  as  follows  :  the  first  for  $425,  due 
April  1,  1875  ;  the  second  for  $615,  due  May  10,  1875  ; 
the  third  for  $1500,  due  May  28,  1875  ;  and  the  fourth 
for  $750,  due  June  10, 1875. 

At  what  date  should  a  single  note  be  made  payable,  to 
be  given  in  exchange  for  the  four  notes  ? 

AVERAGING    ACCOUNTS. 

735.  An  Account  is  a  written  statement  of  debit 
and  credit  transactions,  with  their  respective  dates. 

Debit  means  what  is  owed  by  the  person  with  whom  the  account 
is  kept ;  credit,  what  is  due  to  him  from  the  person  keeping  the 
account. 

736.  To  Average  an  Account  is  to  find,  either 


LVEKAGItfG     ACCOUNTS. 


3?:> 


the  equated  time  of  paying  the  balance,  or  the  cash  balance 
at  any  given  time. 

Each  item  of  a  book  account  should  draw  interest  from  the  time 
it  becomes  due. 


WRITTEN    EXERCISES. 

1.  Find  the  equated  time  of  paying  the  balance 


•737 

of  the  following  account. 

Dr.  William  Sampson. 


Cr. 


1875. 

1875. 

Jan.  11 

To  mdse.  .  .  . 

$750 

Feb.  10 

By  draft  at  60  da. 

$500 

Feb.  1 

"   "  at  3  mo. 

600 

Mar.  3 

"  cash  .  .  . 

700 

Mar.  15 

"   ■'  at  6  mo. 

1500 

Apr.  15 

<<  << 

300 

May  3 

'*      "  at  4  mo. 

900 

Operation  I.    {Method  by  Products.) 


Due. 


Amt.     Days.       Product. 


Jan.   11.      750  x     10 


7500 


Paid. 


Amt. 


Apr.  14.      500  x  103  =     51500 


May  1. 

600  x  120 

a  72000 

Mar.  3.   700  x  61  =  42700 

Sept.  15. 

1500  x  257 

=  385500 

Apr.  15.  300  x  104  =  31200 

"   3. 

900  x  245 

=  220500 

1500        125400 

3750 

685500 

1500 

125400 

2250 

)  560100 

24811,  or  249  da. 
Balance  due  249  da.  from  Jan.  1,  or  Sept.  7. 

Analysis. — Assuming  for  convenience  Jan.  1  as  the  standard 
date,  we  find  as  in  734  the  term  of  credit  of  each  debit  amount ; 
and,  reckoning  from  the  same  date,  the  time  to  each  credit  amount. 
Multiplying  each  amount  by  its  time  in  days,  and  adding  the  debit 
and  credit  products,  we  find  the  number  of  days'  interest  of  $1  due 
to  the  debtor,  and  the  number  of  days'  interest  of  $1  he  has  already 
received.  The  difference,  560100,  shows  the  number  of  days'  inter- 
est of  $1  still  due,  and  as  the  balance  is  $2250,  the  time  must  be 
Y^Vff  of  560100  da.,  or  249  da.  Hence,  the  equated  time  is  249  da. 
from  Jan.  1,  or  Sept.  7. 


376 


PEKCENTAGE, 


Dr. 


Operation  II.    {Method  by  Interest.) 


$750  to  Jan.  11  (from  Jan.  1)= 


10  da.,  int.  at  1%  per  mo.  $2.50 


600 

"  Feb.   1  +  3  mo. 

=  4  mo. 

<  < 

ii 

24.00 

1500 

"  Mar.  15  +  6  mo. 

=  8  mo.  14  da., 

" 

<< 

127.00 

900 

"  May   3  +  4  mo. 

=  8  mo.    2  da., 

«< 

t* 

72.60 

$3750 

$226.10 

Or. 

$500  to  Feb.  10  +  63  da.  =  3  mo.  13  da.,  int.  at  1%  per  mo.  $17.17 
700  "  Mar.   3  =  2  mo.    2  da.,      "  "  14.47 

300  "  Apr.  15  =3  mo.  14  da.,      «  "  10.40 

$1500  $42.04 

$226.10  -  $42.04  =  $184.06,  int.  at  1%  per  mo.  due. 
Int.  of  balance,  $2250,  for  1  mo.,  at  lc/0  =  $22.50. 
Hence,  $184.06  -^  $22.50  =  8.18+  mo.,  or  8  mo.  6  da. 
8  mo.  6  da.  from  Jan.  1,  or  Sept.  7,  Equated  Time. 

In  this  operation,  12  %  per  annum  or  1  %  per  mo.  is  assumed  for 
convenience  ;  since  the  int.  at  1  %  per  mo.  is  as  many  hundredths 
as  there  are  months,  and  one-third  as  many  thousandths  as  there 
are  days.  Thus,  the  int.  of  $249  for  2  mo.  9  da.  is  $498  H  $.747 
=  $5,727(571). 


2.  Find  the  equated  time  of  the  following 
Dr.  William  Simpson. 


Or. 


1874. 

1874. 

Aug.   5 

To  mdse.  at  3  mo. 

$720 

Oct.  10 

By  cash     .    .    . 

$500 

Sept.  10 

"      "      "  2   " 

850 

Dec.  15 

"  draft  at  60  da. 

450 

Nov.    3 

t*      ** 

1200 

"    25 

"  cash     .     .     . 

900 

1875. 

1875. 

Jan.  20 

"  sundr's  at  5  mo. 

620 

Jan.    3 

it     i< 

250 

Rule  1. — I.  Find  the  date  at  which  each  debit  item  is 
(hie,  and  each  credit  item  is  paid  or  due. 

II.  Take  the  first  day  of  the  month  in  the  earliest  date 
on-  either  side  of  the  account  as  a  standard  date,  and  mul~ 


AVERAGING     ACCOUNTS 


377 


tiply  each  sum  due  or  paid  by  the  number  of  days  between 
its  time  and  the  standard  date. 

III.  Add  the  products,  and  their  difference  divided  by 
the  balance  due  will  give  the  number  of  days  between  the 
standard  date  and  the  equaled  time.     Or, 

Rule  2. — Find  the  time  of  each  item  from  the  standard 
date  as  before,  and  compute  the  interest  on  each  at  1%  « 
month.  The  difference  between  the  amount  of  interest  on 
each  side  divided  by  the  interest  of  the  balance  at  \%for 
one  month  will  be  the  equated  time. 

When  the  terms  of  credit  are  long,  Rule  2.  gives  the  shorter 
method. 

3.  Find  the  equated  time  of  the  following,  allowing 

60  da.  credit  on  each  debit  item : 

Dr.  John  Deiscoll.  Cr. 


1877. 

1877. 

June  1 

To  mdse.  .  . 

$950 

Aug.  1 

By  cash   .  . 

$700 

July  6 

tt      it 

300 

Sept.20 

<  <   << 

1000 

Sept.  8 

<<   «< 

1900 

Nov.  1 

tt       a 

1200 

Oct.  20 

tt      « t 

2600 

4.  What  is  the  equated  time  for  the  payment  of  the 
balance  of  the  following  account,  allowing  4  months' 
credit  on  all  the  debit  items  ? 
Dr.  Dodd,  Brown  &  Co.  Cr. 


'   1878. 

1878. 

Jan.  20 

To  mdse.  .  . 

$570 

Feb.  14 

By  mdse.  .  . 

$490 

"  28 

tt          tt 

300 

Mar.  1 

'■*   cash   .  . 

1000 

Feb.  11 

tt          <t 

720 

Apr.  2 

tt      tt 

1800 

"  26 

<t          tt 

835 

Mar.  10 

<<          it 

1150 

"  28 

tt          tt 

900 

Apr.  15 

tt          tt 

475 

378 


PERCENTAGE. 


738.  1.  Find  the  cash  balance  of  the  following  acconnt 
on  the  22d  of  August,  allowing  interest  at  6%  : 

Dr.  George  Hammond.  Or. 


1875. 

1875. 

Mar.  15 

Tomdse  ,at3mo. 

$600 

May  10 

By  cash      .    . 

$300 

Apr.    3 

"      "      "4mo. 

700 

July    1 

<«      <t 

400 

May  10 

"      "      "6  mo. 

1000 

Aug.  15 

«<      «< 

500 

Operation. — By  averaging  the  account,  the  equated  time  for 
paying  the  balance,  $1100,  is  found  to  be  Nov.  4.     (734.) 

True  present  worth  of  $1100  for  74  da.  (from  Aug.  22  to  Nov.  4) 
is  $1086.60,  or  cash  balance  Aug.  22. 
Or,  by  Interest  Method,  as  follows  : 

Br. 
Int.   of    $600,  from  June  15  to  Aug.  22,. 
"      "      700,     "     Aug.    3  " 

(Jr. 
Int.  of  $1000,  from  Aug.  22  to  Nov.  10, 
"       "       300,     "     May  10  "  Aug.  22, 
"       "      400,     "     July    1" 
"      "      500,     "     Aug.  15" 


Balance  of  interest  due  Hammond,        $13.38 
$1100  -  $13.38  =  $1086.62,  Cash  Balance,  Aug.  22. 

Analysis. — Charge  Hammond  with  interest  on  each  debit  item 
from  the  time  it  is  due  to  date  of  settlement,  and  credit  him  witb^ 
interest  on  each  sum  paid  from  the  date  of  payment  to  date  of  set- 
tlement, also  on  each  debit  item  which  becomes  due  after  the  date 
of  settlement.  Hence,  he  is  entitled  to  interest  on  $1000  from 
Aug.  22  to  Nov.  10.  As  the  balance  of  interest  is  in  favor  of  Ham- 
mond, it  must  be  deducted  from  the  balance  of  the  account,  to  ob- 
tain the  cash  balance.  There  is  a  slight  difference  in  the  results, 
but  the  interest  method  is  the  more  accurate.  By  the  use  of  Inter, 
est  Tables,  it  is  also  the  shorter  of  the  two  methods. 


68  da, 
19  da., 

$6.71  (574. 
2.19 

$8.90 

80  da., 

[04  da, 

52  da., 

7  da. 

$13.15 

513 

3.42 

58 

$22.28 
8.90 

AVEKAGING     ACCOUNTS 


379 


Rule  1. — I.  Average  the  account,  and  find  the  equated 
time  of  payment  of  the  balance. 

II.  If  the  date  of  settlement  is  prior  to  the  equated  time, 
find  the  present  worth  of  the  balance  of  account  for  the 
cash  balance  ;  if  later,  find  the  interest  of  the  balance  of 
account  for  the  intervening  time,  and  add  it  to  find  the 
cash  balance.    Or, 

Eule  2. — Find  the  interest  on  each  debit  and  credit 
item,  from  the  time  it  is  due  or  paid  to  the  date  of  settle- 
ment, placing  on  the  same  side  of  the  account  the  interest 
on  each  item  due  prior  to  the  date  of  settlement,  and  on  the 
opposite  side  the  interest  on  each  item  due  after  the  date 
of  settlement.  If  the  balance  of  interest  is  on  the  same  side 
as  the  balance  of  the  account,  add  it,  if  on  the  other  side 
subtract  it ;  and  the  result  xoill  be  the  cash  balance  at  the 
date  of  settlement. 

2.  I  owe  $1500  due  May  1,  and  $750  due  Aug.  15.  If 
I  give  my  note  at  30  da.  for  $450,  June  1,  and  pay  $370 
in  cash  July  15,  what  is  the  equated  time  for  paying  the 
balance  ;  and  what  would  be  due  in  cash  Dec.  10,  allow- 
ing interest  at  7%  ? 

3.  When  is  the  balance  of  the  following  account  due 
1  per  average  ? 

Dr.  0.  B.  Timpsok.  Or. 


1875. 

Aug.  10 

To  mdse.  @  60  da.  . 

Sept.  5 

u      "      @30da.  . 

Nov.    1 

"      M      @90da.  . 

Dec.    5 

"      "      @^0da.  . 

I 

$751.33 
425.00! 
927.83  j 

1200.00 


1875. 
Oct.    3 
Nov.  15 
Dec.  20 


By  cash .  .  .  . 
"  note  @.  90  da. 
"   cash     .    .    . 


$300.00 
450.00 
500.00 


380 


PERCENTAGE 


4.  What  is  the  cash  balance  of  the  above  account  Jan.  1, 
1876,  allowing  interest  at  10$  ? 

5.  Find  the  equated  time,  and  cash  balance  July  1,  of 
the  following,  allowing  1%  interest : 

Dr.  Thomas  Smith.  Cr. 


Jan.    4 

To  mdse.  @  4  mo. 

$1600 

Feb.    1 

By  mdse.  @  4  mo. 

$500 

"      6 

"       "      @  3  mo. 

1500 

Mar.   2 

"  cash    .     .     . 

2000 

Apr.  10 

*      "      @60da. 

3000 

"     25 

(t       u 

3150 

"    28 

*      *      @30da. 

2500 

Apr.  16 

"■    "       ... 

800 

6.  Average  the  following  account,  and  find  for  what 
amount  a  note  at  60  days  should  be  given  Aug.  1,  to  pay 
the  balance,  interest  at  6< 


Dr. 


Orson  Hinmak. 


Cr. 


1875. 

1875. 

Apr.  2 

To  charges 

$87.25 

Feb.  25 

By  mdse.  @  8  mo 

$600 

May  15 

<<        << 

35.75 

Mar.    3 

"       "      @6  " 

300 

|Apr.    1 

"      «      @6  " 

500 

739.  1.  Average  the  following  Account  Sales,  and  find 
when  the  net  proceeds  are  due.     (543.) 

Account  Sales  of  1200  Ills,  of  flour  received  from 
Smithy  Tyler  &  Co.,  Cincinnati. 


Date. 

Buyer. 

Quantity . 

Price. 

Amount.    « 

1876. 
May  1 
June  5 

«'  15 
July   1 

J.  Brooke 
W.  Long 
A.  Bruce 
W.  Case 

300  bbl. 
450   " 
250   " 
200  " 

%  $5.50,  3  mo. 
@    6.20,  4  mo. 
@    6.50,  6  mo. 
@    5.75,  2  mo. 

$1650.00 
2790.00 
1625.00 
1150.00 

$7215.00 


AVERAGING     ACCOUNTS.  381 

CHAKGES. 

Apr.  28.    Freight $674.50 

"      "      Cartage 37.50 

May    1.     Storage 191.00 

Commission  on  $7215,  ®2£%     .    .    .    162.34 

Total  charges $1065.34 

Net  proceeds  due  per  average $6149.66 

OPERATION. 

Average  of  sales,  found  by  the  method  of  Equation  of  Payments, 
Oct.  lt  which  is  the  date  at  which  the  commission  is  due. 

Average  of  charges,  including  commission  (Oct.  1),  May  22. 
Equated  time  of  $7215  due  Oct.  1,  and  $1065.34  due  May  22,  Oct  24, 
date  when  the  net  proceeds  are  due. 

Kule. — I.  Average  the  sales  alone,  and  the  result  will 
be  the  date  to  be  given  to  the  commission  and  guaranty. 

II.  Make  the  sales  the  credits  and  the  charges  the  debits, 
and  find  the  equated  time  for  paying  the  balance. 

2.  Make  an  account  sales,  and  find  the  net  proceeds 
and  the  time  the  balance  is  due  : 

Wm.  Brown,  of  N.  York,  sold  on  acct.  of  J.  Berry,  of  Chi- 
cago, June  1,  350  bu.  Winter  Wheat,  @  $1.35,  at  60  da. ; 
June  15,  275  bu.  Spring  Wheat,  @  $1. 75,  at  90  da, ;  July  3, 
1260  bu.  Indian  Corn,  @  $.79,  at  6  mo.;  and  July  10, 
375  bu.  Eye,  @  $1.02,  at  3  mo.  Paid  freight,  May  28, 
$567.50 ;  cartage,  May  30,  $22.50  ;  insurance,  June  5, 
$56.25  ;  and  charged  com.  at  3\%,  and  lj-%  for  guaranty. 

3.  Sold  on  account  of  Brown,  Sampson  &  Co.,  at  6 
mo. :  Oct.  1,  1874,  13  hhd.  sugar,  averaging  1520  lb.,  @ 
$.12|  ;  Oct.  5,  15  chests  Hyson  Tea,  each  95  lb.,  @  $1.05. 
Paid  charges :  Oct.  3,  Insurance,  $85  ;  Oct.  10,  Cooper- 
age, etc.,  $24.50  ;  Oct.  20,  Cartage,  $125.  Charged  com- 
mission and  guaranty,  ty%.  Make  an  account  sales,  and 
find  the  equated  time  for  paying  the  net  proceeds. 


382 


PERCENTAGE 


740. 


SYNOPSIS    FOR    REVIEW. 

1.  Exchange.  2.  Domestic  Exchange.  3.  For* 
eign  Exchange.  &.  Bill  of  Exchange.  5. 
Set  of  Exchange.  6.  Sight  Draff  or  Bill. 
7.  Time  Draft  or  Bill.  8.  Buyer  or  Re- 
mitter. 9.  Acceptance.  10.  Par  of  Exchange. 
11.  Course  or  Bate  of  Exchange. 
2.  Forms.    1.  A  Sight  Draft.    2.  Set  of  Exchange. 


1.  Defs. 


3.  Inland 
Exch 


d  j  1.  700.  ) 
.    (  2.  70i.  J 


To  find 


\  Cost  of Draft.  Formula. 
|  Face  of  Draft. 


4.  Foreign 
Exch'ge. 


5.  Arbitra- 
tion of 
Exch'ge. 


Custom- 
house 
Business. 


23.   Equation 
op  Paym'ts. 


24.  Averaging 
Accounts. 


j  1.  Money  of  Account. 
(2. 
.an| 

':( 


To  find 


j  Cost  of  Bill. 
\  Face  of  Bill. 


1.  Defs. 


Sterling  Bills,  or  Exchange 
2.  Exchange  with  Europe — how  effected. 
8.  706. 
4.  707.  . 

1.  Arbitration  of  Exchange. 

2.  Simple  Arbitration. 

I  3.  Compound  Arbitration. 
2.  Rule,  I,  II,  III,  IV 

1.  Custom  House.  2.  Port  of 
Entry.  3.  Collector.  4. 
Clearance.  5.  Manifest.  6. 
Duties  or  Customs.  7.  Tariff. 
8.  Specific  Duty.  9.  Ad  Val- 
orem Duty.  10.  Gross  Wght. 
11.  Net  Weight- 
To  find  the  Duty. 
1.  Equation  of  Payments.  2. 
Equated  Time.  3.  Term  of 
Credit.  4.  Average  Term  of 
Credit. 

2.  Principle. 

3.  733.    Rule,  I,  II. 

4.  734.     Rule,  I,  II,  III.     Proof. 

1.  Defs.    1.  Account.  2.  To  Average  an  Acct 

2.  737.    Rule  1,  I,  II,  III.    Rule  2. 

3.  738.    Rule  1, 1,  II.    Rule  2. 

4.  739.    Rule,  I,  II. 


1.  Defs.  J 


2.  726. 


1.  Defs. 


t>RAE    exercises. 

741.  1.  A  father  is  30  years  old,  and  his  son  6  ;  how 
many  times  as  old  as  the  son  is  the  father  ? 

2.  30  are  how  many  times  6  ?    30  '-*-  6  =  I 

3.  What  part  of  .$30  are  $G  ?    Of  20  cents  are  5  cents  ? 

4.  What  is  the  relation  of  8  to  2  ?     Of  40  rd.  to  4  rd.  ? 

5.  What  relation  has  12  to  3  ?     GO  lb.  to  20  lb.  ? 
Compare  the  following,  and  give  their  relative  yalnes: 


G.   75  with  5. 

7.  25  with  6J. 

8.  1  with  7. 


9.     \  with  7. 

10.  24  with  3f 

11.  .9  with  .3. 


12.  $.6  with  $.2. 

13.  .42  with  .3. 

14.  f  with  }. 


DEFINITIONS. 

742.  JRatio  is  the  relation  between  two  numbers  of 
the  same  unit  yalue,  expressed  by  the  quotient  of  the  first 
divided  by  the  second.  Thus  the  ratio  of  12  to  4  is 
12  -~  4  =  3. 

743.  The  Sign  of  ratio  is  the  colon  ( : ),  or  the  sign 

of  division  with  the  line  omitted. 

Thus,  the  ratio  of  9  to  3  is  expressed  9:3,  or  9-5-3,  or  in  the  form 
of  a  fraction  f ,  and  is  read,  the  ratio  of  9  to  3,  or  9  divided  by  3. 

744.  The  Terms  of  a  ratio  are  the  two  numbers 
compared. 

745.  The  Antecedent  is  the  first  term,  or  dividend. 

746.  The  Consequent  is  the  second  term,  or  divisor. 


384  RATIO. 

74:7.  The  Value  of  a  ratio  is  the  quotient  of  the  antece- 
dent divided  by  the  consequent,  and  is  an  abstract  number. 

Thus,  in  the  ratio  $18  :  $6,  $18  and  $6  are  the  terms  of  the  ratio  ; 
$18  is  the  antecedent ;  $6  is  the  consequent ;  and  3,  the  quotient  of 
$18-5-  $0,  is  the  value  of  the  ratio. 

748.  A  Simple  Matio  is  the  ratio  of  two  numbers  ; 
as  10  :  5. 

749.  A  Compound  Matio  is  the  ratio  of  the 
products  of  the  corresponding  terms  of  two  or  more  sim- 
ple ratios. 

Thus  the  ratio  compounded  of  the  simple  ratios, 

*)&*** expressed I  or, (8 *  t' ■  49x: n  \  =72 : 48 ; 

Or,  f  *  ft  =  ♦  =  8  :> 

When  the  multiplication  is  performed  the  result  is  a  simple  ratio. 

750.  The  Reciprocal  of  a  ratio  is  1  divided  by  the 
ratio  (196),  or  it  is  the  consequent  divided  by  the  ante- 
cedent. Thus  the  ratio  of  8  to  9  is  8  :  9,  or  f,  and  its 
reciprocal  is  f . 

The  ratio  of  two  fractions  is  obtained  by  reducing  them  to  a 
common  denominator,  when  they  are  to  each  other  as  their  nume- 
rators (241). 

If  the  terms  of  a  ratio  are  denominate  numbers,  they  must  be 
reduced  to  the  same  unit  value. 

751.  From  the  preceding  definitions  and  illustrations 
are  deduced  the  following 

Foemulas. — 1.   The  Ratio  =  Antecedent -r-  Consequent, 

2.  The  Consequent  =  Antecedent-- Ratio. 

3.  The  Antecedent  as  Consequent  x  Ratio. 


RATIO.  385 

752*  Since  the  antecedent  is  a  dividend,  and  the  con- 
sequent a  divisor,  any  change  in  either  or  both  of  the 
terms  of  a  ratio  will  affect  its  value  according  to  the  laws 
of  division  or  of  fractions  (200),  which  laws  become  the 

General  Principles  of  Eatio. 

1.  Multiplying  me  antecedent,  or  \  MuWlUstlleratio, 
Dividing  the  consequent,  ) 

2.  Dividing  the  antecedent,  or       )  ^.^  fc  ^ 
Multiplying  the  consequent,       ) 

3.  Multiplying  or  dividing  both  )  -.  ,  ,, 

antecedent  and  consequent  by  r         .. 
the  same  number,  ) 

753.  These  principles  may  be  embraced  in  one 

GENERAL  LAW. 

A  change  in  the  antecedent  produces  a  like  change  in 
the  ratio ;  but  a  change  in  the  consequent  produces  an 
opposite  change  in  the  ratio. 

EXERCISES. 

754.  1.  Express  the  ratio  of  11  to  4  ;  of  16  to  2  ;  of  20 
to  6| ;  of  $36  to  $12  ;  of  9  lb.  to  27  lb.  ;  of  4$  bu.  to  9  bu. 

2.  Can  you  express  the  ratio  between  $15  and  5  lb.  ? 
The  reason  ? 

3.  Indicate  the  ratio  of  18  to  20  in  two  forms.  What 
are  the  terms  of  the  ratio  ?  The  antecedent  f  The  con- 
sequent?   The  hind  of  ratio  ?    The  value  of  the  ratio. 

In  like  manner  express,  analyze,  and  give  the  value, 

4.  Of  80  to  120  ;  of  12£  to  37| ;  of  16-J-  to  f . 

5.  Of  5.3  to  1.3;  of  |  to  A;  of  §$££ 

17 


386  RATIO. 

6.  The  antecedents  of  a  ratio  are  7  and  10,  and  the 
consequents,  5  and  4.    What  is  the  value  of  the  ratio  ? 

7.  The  first  terms  of  a  ratio  are  18,  12,  and  30,  the 
second,  54,  6,  and  15.  What  is  the  kind  of  ratio  ?  Ex- 
press in  three  forms.     Find  its  value  in  the  lowest  terms. 

Solve,  and  state  the  formula  applied  to  the  following  : 

8.  The  consequent  is  3£,  the  antecedent  ff ;  what  is 
the  ratio  ? 

9.  The  antecedent  is  60,  the  ratio  7  ;  what  is  the  con- 
sequent ? 

10.  The  consequent  is  I6.12J,  the  ratio  ff  ;  what  is 
the  antecedent  ? 

11.  The  ratio  is  2f,  the  antecedent  J  of  £  ;  what  is  the 
consequent  ? 

12.  The  ratio  is  6,  the  consequent  1  wk.  3  da.  12  hr. ; 
what  is  the  antecedent  ? 

13.  Express  the  ratio  of  120  to  80,  and  give  its  value 
in  the  lowest  terms. 

14.  Make  such  changes  in  the  last  example  as  will 
illustrate  Prin.  1. 

15.  With  the  same  example,  illustrate  Prin.  2. 

16.  Illustrate  by  the  same  example  Prin.  3. 

17.  Find  the  reciprocal  of  the  ratio  of  75  to  15. 

18.  Find  the  reciprocal  of  the  ratio  of  2  qt.  1  pt.  to 
4  gal.  1  qt.  1  pt. 

What  is  the  ratio 

19.  Of  40  bu.  4.5  pk.  to  25  bu.  2  pk.  1  qt. 

20.  Of  6  A.  110. P.  to  10  A.  60  P. 

21.  Of  25  lb.  11  oz.  4  pwt.  to  19  lb.  5  oz.  8  pwt 

m  of  m  to  tgi. 


OBAI,      EXERCISES. 

155.  1.  What  is  the  ratio  of  4  to  2  ?    Of  6  to  1  ?    Of 

14  to  7  ?     Of  21  to  3  ? 

2.  Find  two  numbers  that  have  the  same  quotient  as 
8  -=-  2.     As  27  -f-  3.    As  16  -f-  4.     As  30  -f-  6.     As  4^-f 

3.  Express  in  the  form  of  a  fraction  the  ratio  of  26  to 
13.     Of  32  to  8. 

4.  Express  in  both  forms  the  ratio  of  two  other  num- 
bers equal  to  the  ratio  of  10  to  2.    Of  15  to  5.    Of  12  to  3. 

5.  If  4  stamps  cost  12  cents,  what  will  20  stamps  cost 
at  the  same  rate  ? 

6.  What  number  divided  by  12,  gives  the  same  quo- 
tient as  20  -r-  4  ? 

7.  What  number  has  the  same  ratio  to  12,  that  20  has 
to  4? 

8.  To  what  number  has  48  the  same  ratio  that  80  has 
to  5?    That  24  has  to  3  ? 

9.  The  ratio  of  20  to  5  is  the  same  as  the  ratio  of  what 
number  to  4  ?    To  6  ?     To  5£  ?    To  6 J-  ? 

10.  The  ratio  of  45  to  9  is  the  same  as  the  ratio  of  15 
to  what  number  ?     Of  30  to  what  number  ? 

11.  28  is  to  7  as  8  is  to  what  number  ? 

12.  56  is  to  8  as  what  number  is  to  5  ? 

13.  63  —  what  number  equals  the  ratio  of  36  to  4  ? 


388  PROPORTION 


DEFINITIONS. 

756.  A  Proportion  is  an  equation  in  which  each 
member  is  a  ratio  ;  or  it  is  an  equality  of  ratios. 

757.  The  equality  of  the  two  ratios  may  be  indicated 
by  the  sign  =  or  by  the  double  colon  : : 

Thus,  we  may  indicate  that  the  ratio  of  8  to  4  is  equal  to  that  of 
6  to  3,  in  any  of  the  following  ways  : 

8:4  =  6:3,  8:4::6:3,  ?  =  &  8  h- 4  =  6 -^  3. 

4      6 

This  proportion,  in  any  of  its  forms,  is  read,  The  ratio  of  8  to  If.  is 
equal  to  the  ratio  of  6  to  3,  or,  8  is  to  4  as  6  is  to  3. 

758.  Since  each  ratio  consists  of  two  terms,  every  pro- 
portion must  consist  of  at  least  four  terms.  Each  ratio  is 
called  a  Couplet,  and  each  term  is  called  a  Proportional. 

759.  The  Antecedents  of  a  proportion  are  the  first 
and  third  terms,  that  is,  the  antecedents  of  its  ratios. 

760.  The  Consequents  are  the  second  and  fourth 
terms,  or  the  consequents  of  its  ratios. 

761.  The  Extremes  are  the  first  and  fourth  terms. 

762.  The  Means  are  the  second  and  third  terms. 

In  tho  proportion  8  :  4  : :  10  :  5,  8,  4,  10,  and  5  are  the  propor- 
tionals; 8  :  4  is  the  first  couplet,  10  :  5  the  second  couplet ;  8  and  10 
are  the  antecedents,  4  and  5  are  the  consequents;  8  and  5  are  the  ex- 
tremes, 4  and  10  are  the  means. 

TJiree  numbers  are  proportional,  when  the  ratio  of  the  first  to  the 
second  is  equal  to  the  ratio  of  the  second  to  the  third.  Thus  the 
numbers  4,  10,  and  25  are  proportional,  since  4 :  10  =  10  :  25,  the 
ratio  of  each  couplet  being  f. 

When  three  numbers  are  proportional,  the  second  term  is  called 
a  Mean  Proportional  between  the  other  two. 


PROPORTION.  389 

The  proportion    8  .  4  : :  10  :  5    may  be  expressed  thus,  f  =  -^ 

(757).    Reducing  these  fractions  to  equivalent  ones  having  a  coni- 

8x5      10  x  4 
mon  denominator,  — —  =  . 

Since  these  fractions  are  equal,  and  have  a  common  denominator, 
their  numerators  are  equal,  or  8  x  5  =  10  x  4. 

763.  Principles.— 1.  The  product  of  the  extremes  of 
a  proportion  is  equal  to  the  product  of  the  means. 

2.  The  product  of  the  extremes  divided  by  either  mean 
will  give  the  other  mean. 

3.  Tlie  product  of  the  means  divided  by  either  extreme 
will  give  the  other  extreme. 

EXERCISES. 

764.  1.  The  ratio  of  4  to  10  is  equal  to  the  ratio  of  6 
to  15.     Express  the  proportion  in  all  its  forms  (757). 

Drill  Exercise. — How  many  terms  has  a  proportion  ?  What  are 
they  called  ?    How  many  ratios  ?     What  are  they  called  ? 

Name  the  proportionals  in  example  1 ;  the  couplets  ;  the  ante- 
cedents ;  the  consequents  ;  the  extremes  ;  the  means.  What  is  the 
product  of  the  extremes  ?  Of  the  meaus  ?  What  is  the  dividend 
of  the  first  ratio  ?  The  divisor  of  the  second  ratio  ?  The  divisor 
of  the  first  ratio  ?  The  dividend  of  the  second  ratio  ?  In  the  frac- 
tional form  what  are  the  numerators  ?    The  denominators  ? 

2.  The  ratio  of  6  to  15  equals  the  ratio  of  8  to  20. 

3.  The  ratio  of  4 £  to  18  equals  the  ratio  of  6  to  24. 
Change  to  the  form  of  equations  by  Prin.  1  : 


4.  12  :  1728  : :  1  :  144. 

5.  2|  :  1?  : :  20  :  143&. 


6.  27.03  :  9.01  : :  16.05  :  5.35. 

8.  The  extremes  are  15  and  48,  and  one  of  the  means 
is  10.     Find  the  other  mean. 

9.  The  means  are  25  and  75,  and  one  of  the  extremes 
is  12£.     Find  the  other  extreme. 


390 


PKOPOKTION. 


The  required  or  omitted  term  in  a  proportion,  or  in  an  operation, 
will  hereafter  be  represented  by  x. 

Find  the  term  omitted  in  each  of  the  following  pro- 
portions : 

17.  4Jyd.:zyd.::$9£:  $27.25. 

18.  x:  9.01::  16.05  :  5.35. 

19.  |  yd. :  x  yd. : :  $| :  $59.0625. 

20.  ^:|::^:|. 

21.  a:38i::8£:76f 

22.  7.5  :18::zoz.  :  7TV  oz. 


11.  8:  52::  20  :  #. 

12.  12:a::  1  :  144. 

13.  x:  20::  120:  50. 

14.  $80:  Hi:  x:  4. 

15.  2.5  :  62.5  ::5:x. 

16.  $175.35  :tx::t:$. 


SIMPLE    PKOPORTION. 

765.  A  Simple  Proportion  is  an  expression  of 

equality  between  two  simple  ratios.     It  is  used  to  solve 

problems  of  which  three  terms  are  given,  and  the  fourth 

is  required. 

Of  the  three  given  numbers,  two  must  always  be  of  the  same 
kind  ;  and  the  third,  of  the  same  kind  as  the  required  term. 

766.  A   Statement  is   the  arrangement  of  these 
terms  in  the  form  of  a  proportion. 

WRITTEN    EXERCISES. 

767.  1.  If  4  tons  of  coal  cost  $24,  what  will  be  the 
cost  of  12  tons  at  the  same  rate  ? 


STATEMENT. 

4T.  :12  T.  ::  $24:  %x 

OPERATION. 

$72 


12  x  24-^-4 

Or  By  Cancellation 
12  x  JM6 


%x  = 


$72 


Analysis.— Since  4  tons  and  12 
tons  have  the  same  unit  value,  they 
can  be  compared,  and  will  form  one 
couplet  of  the  proportion. 

For  the  same  reason  $24  the  cost 
of  4  tons,  and  $x  the  cost  of  12  tons, 
will  form  the  other  couplet. 

Then  by  Prin.  3,  $x=  24  x  12  h-4 

=  m 


PROPORTION.  391 

Proof. — 4  x  72=12  x  24.  (763,  Prin.  1.)  In  practice,  that  number 
vhich  is  of  the  same  unit  value  as  the  required  term,  is  generally 
made  the  antecedent  of  the  second  couplet  or  third  term  of  the  pro- 
portion, and  the  required  term,  x,the  fourth  term.  The  terms  of  the 
first  couplet  are  so  arranged  as  to  have  the  same  ratio  to  each  other, 
as  the  terms  of  the  second  couplet  have  to  each  other,  which  is 
easily  determined  by  inspection.  The  product  of  the  means  12  and 
24,  divided  by  the  given  extreme  4,  gives  the  other  extreme,  or 
required  term,  $72.    (763,  Prin.  3.) 

Brill  exercises  like  the  following,  will  soon  make  the  pupil 
familiar  with  the  principles  and  operations  of  proportion. 

2.  If  4  horses  eat  12  bushels  of  oats  in  a  given  time, 
how  many  bushels  will  20  horses  eat  in  the  same  time  ? 

In  this  example,  what  two  numbers  have  the  same  unit  value  ? 
What  do  they  form  ?  What  is  the  denomination  of  the  third  term  ? 
Of  the  required  term  ?  What  is  the  antecedent  of  the  second 
couplet  ?  From  the  conditions  of  the  question,  is  the  consequent 
of  the  second  couplet  or  required  term,  greater  or  less  than  the 
antecedent  ?  If  greater,  how  must  the  antecedent  and  consequent 
of  the  first  couplet  compare  with  each  other  ?  If  less,  how  com- 
pare ?  What  is  the  ratio  of  the  first  couplet  ?  Why  not  20  to  4  ? 
Make  the  statement.    How  is  the  required  term  found  ? 

3.  If  96  cords  of  wood  cost  $240,  what  will  40  cords  cost  ? 

4.  If  20  lb.  of  sugar  cost  $1.80,  find  the  cost  of  45  lb. 

5.  If  18  bu.  of  wheat  make  4  barrels  of  flour,  how  many 
barrels  will  200  bu.  make^? 

Kule. — I.  Make  the  statement  so  that  tivo  of  the  given 
numbers  which  are  of  the  same  unit  value,  shall  form  the 
first  couplet  of  the  proportion,  and  have  a  ratio  equal  to 
the  ratio  of  the  third  given  term  to  the  required  term. 

II.  Divide  the  product  of  the  means  by  the  given  extreme, 
and  the  quotient  will  be  the  number  required. 


392  PROPOETIOJS'. 

CAUSE    AND    EFFECT. 

768.  The  terms  of  a  proportion  have  not  only  the 
relations  of  magnitude,  but  also  the  relations  of  cause 
ani  effect. 

Every  problem  in  proportion  may  be  considered  as  a 
comparison  of  two  causes  and  two  effects. 

Thus,  if  4  tons  as  a  cause  will  bring  when  sold,  $24  as  an  effect, 
12  tons  as  a  cause  will  bring  $72  as  an  effect.  Or,  if  6  horses  as  a 
cause  draw  10  tons  as  an  effect,  9  horses  as  a.  cause  will  draw  15 
tons  as  an 


769.  Since  like  causes  produce  like  effects,  the  ratio 
of  two  like  causes  equals  the  ratio  of  two  like  effects  pro- 
duced by  these  causes.     Hence, 

1st  cause  :  2d  cause  : :  1st  effect :  2d  effect. 

WRITTEN    EXERCISES. 

770.  1.  If  8  men  earn  $32  in  one  week,  how  much  will 
15  men  earn  at  the  same  rate,  in  the  same  time  ? 

STATEMENT.  ANALYSIS —In  this  ex- 

ist cause.         2d  cause.       1st  effect.  2d  effect,     ample  an  effect  is  required. 
8  men   :   15  men   ::    $32   :   $2        The  first  cause  is  8  men, 

the  second  cause  15  men, 

OPERATION.  ,      .  . 

and  since  they  are  like 


$#  —  lo  X  <J«-rb  =  $bU  causes  they  can  be  com 

pared. 
The  effect  of  the  first  cause  is  $32  earned,  the  effect  of  the  second 
cause  is  $#  earned,  or  the  required  term.  Since  like  effects  have 
the  same  ratio  as  their  causes  (769),  the  causes  may  form  the 
first  couplet,  and  the  effects  the  second  couplet  of  the  proportion 
The  required  term  is  readily  obtained  by  (763,  3). 

2.  If  20  bushels  of  wheat  produce  6  barrels  of  flour, 
how  many  bushels  will  be  required  to  produce  24  barrels  ? 


PROPOETIOK.  393 

STATEMENT.  ANALYSIS.— In  this  ex- 

ist cause.     2d  cause.     1st  effect.     2d  effect.      ample  a  cause  is  required. 
20  bu.  :  x\)Vl  ::  6  bbl.  :  24  bbl.        The  first  cause  is  20  bu., 

the  second  cause  is  x  bu. 

OPERATION.  ..  .      _  . 

or  the  required  term. 


X  bu.  =  20  X  24  -f-  6  =  80  bu.  The  effect  of  the  first 

cause  is  G  bbl.  of  flour, 
the  effect  of  the  second  cause  is  24  bbl.  of  flour.  Since  like  causes 
have  the  same  ratio  as  their  effects  (769),  the  statement  is  made 
as  in  Ex.  1,  and  the  required  term  found  by  (763,  2). 

3.  If  5  horses  consume  10  tons  of  hay  in  8  mo.,  how 

many  horses  will  consume  18  tons  in  the  same  time  ? 

Drill  Exercise. — In  this  example,  what  is  the  first  cause  ?  The 
second  cause?  The  first  effect?  The  second  effect?  Is  the  re- 
quired term  a  cause  or  an  effect  ?  A  mean  or  an  extreme  ?  What 
is  the  first  couplet?  What,  the  second?  Make  the  statement. 
How  is  the  required  term  found  ? 

4.  If  8  yards  of  cloth  cost  $6,  how  many  yards  can  be 
bought  for  $75  ? 

5.  How  many  men  will  be  required  to  build  32  rods  of 
wall  in  the  same  time  that  5  men  can  build  10  rods  ? 

Rule. — I.  Arrange  the  terms  in  the  statement  so  that 
the  ratio  of  the  causes  which  form  the  first  couplet,  shall 
equal  the  ratio  of  the  effects  which  form  the  second  couplet, 
putting  x  in  the  place  of  the  required  term. 

II.  If  the  required  term  be  an  extreme,  divide  the  pro- 
duct of  the  means  by  the  given  extreme ;  if  the  required 
term  be  a  mean,  divide  the  product  of  the  extremes  by  the 
given  mean. 

To  shorten  the  operation,  equal  factors  in  the  first  and  second,  or 
in  the  first  and  third  terms  may  be  canceled. 

Solve  the  following  by  either  of  the  foregoing  methods  : 

6.  If  5  sheep  can  be  bought  for  $20.75,  how  many 
sheep  can  be  bought  for  $398.40  ? 


394  PROPORTION. 

7.  When  10  barrels  of  flour  cost  $112.50,  what  will  be 
the  cost  of  476  barrels  of  flour  ? 

8.  If  a  railroad  train  run  30  miles  in  50  min.,  in  what 
time  will  it  run  260  miles  ? 

9.  How  many  bushels  of  peaches  can  be  purchased  for 
$454.40,  if  8  bushels  cost  $10.24  ? 

10.  If  a  horse  travel  12  miles  in  1  hr.  36  min.,  how  far, 
at  the  same  rate,  will  he  travel  in  15  hours  ? 

11.  How  many  days  will  12  men  require  to  do  a  piece 
of  work,  that  95  men  can  do  in  7-J-  days  ? 

12.  If  f  of  an  acre  of  land  cost  $60,  what  will  45  J  acres 
cost? 

13.  At  the  rate  of  72  yards  for  £44  16s.,  how  many 
yards  of  cloth  can  be  bought  for  £5  12s.  ? 

14.  If  |  of  a  barrel  of  cider  cost  $1^,  what  is  the  cost 
of  -f  of  a  barrel  ? 

15.  If  the  annual  rent  of  35  A.  90  P.  is  $284.50,  how 
much  land  can  be  rented  for  $374.70  ? 

16.  What  will  87.5  yd.  of  cloth  cost,  if  If  yd.  cost  $1.26  ? 

17.  If  by  selling  $5000  worth  of  dry  goods,  a  merchant 
gains  $456.25,  what  amount  must  he  sell  to  gain  $1000  ? 

18.  Bought  coal  at  $4.48  per  long  ton,  and  sold  it  at 
$7.25  per  short  ton.     What  was  the  gain  per  ton  ? 

19.  What  will  be  the  cost  of  a  pile  of  wood  80  ft.  long, 
4  ft.  wide,  4  ft.  high,  if  a  pile  18  ft.  long,  4  ft.  wide,  6  ft. 
high  cost  $30.24? 

20.  If  36  bu.  of  wheat  are  bought  for  $44.50,  and  sold 
for  $53.50,  what  is  gained  on  480  bu.  at  the  same  rate  ? 

21.  If  a  business  yield  $700  net  profits  in  1  yr.  8  mo.,  in 
what  time  will  the  same  business  yield  $1050  profits  ? 


PKOPOKTIOX. 


395 


COMPOUND    PROPORTION. 

771.  A  Compound  Proportion  is  an  expression 
of  equality  between  two  ratios,  one  or  both  of  which  are 
compound. . 

All  the  terms  of  every  problem  in  compound  proportion  appear 
in  couplets,  except  one,  and  this  is  always  of  the  same  unit  value  as 
the  required  term. 

The  order  of  the  ratios,  and  of  the  terms  composing  the  ratios,  is 
the  same  as  in  simple  proportion. 


WRITTEN     EXERCISES. 

K12.  1.  If  18  men  build  126  rd.  of  wall  in  60  da., 
working  10  hr.  a  day,  how  many  rods  will  6  men  build  in 
110  da.,  working  12  hr.  a  day  ? 


STATEMENT. 

18  men  :  6  men  ) 
60  days  :  110  days  (■ 
10  hours  :  12  hours  ) 


Or, 


rods. 

126 


rods. 
:  it- 


rods. 

X  — 


OPERATION. 
11  42 

1^X00X10 

*       5 


10 


^=92|rd. 


11011 

n 

12042 
462 
92f=zrd. 


Analysis. — All  the  terms  in  this  example  appear  in  couplets,  ex- 
cept 126  rods,  which  is  of  the  same  unit  value  as  the  required  term, 
and  is  made  the  third  term  of  the  proportion,  and  x  rods,  the  fourth. 

The  required  number  of  rods  depends  upon  Preconditions  :  1st, 
the  number  of  men  employed  ;  2d,  the  number  of  days  they  work  ; 
and  3d,  the  number  of  hours  they  work  each  day. 

Consider  each  condition  separately,  and  arrange  the  terms  of  the 
same  unit  value  in  couplets,  and  make  the  statement  as  in  simple 
proportion  (767).    Then  find  the  required  term  by  (763,  3). 


396  PROPORTION. 

2.  If  20  horses  consume  36  tons  of  hay  in  9  mo.,  how 
many  tons  will  12  horses  consume  in  18  months  ? 

Drill  Exercise. — In  this  example,  what  is  the  denomination  of 
the  required  term  ?  What  given  number  has  the  same  unit  value  ? 
What  will  be  the  third  term  of  the  proportion  ?  The  fourth  ? 
How  many  couplets  are  there  ?  Name  them.  What  kind  of  a  ratio 
do  they  form  ?  How  is  the  antecedent  and  consequent  of  each 
couplet  determined  ?  How  is  a  compound  ratio  reduced  to  a  simple 
one  ?  Make  the  statement.  Is  the  required  term  a  mean  or  an 
extreme?    How  is  it  found ?    (763,3.) 

3.  If  $320  will  pay  the  board  of  4  persons  for  8  weeks, 
for  how  many  weeks  will  $800  pay  the  hoard  of  15 
persons  ? 

4.  If  a  man  walk  192  miles  in  6  days,  walking  8  hr.  a 
day,  how  far  can  he  walk  in  18  days,  walking  6  hr.  a  day  ? 

5.  If  6  laborers  can  dig  a  ditch  34  yards  long  in  10 
days,  how  many  days  will  20  laborers  require  to  dig  a 
ditch  170  yards  long? 

Kule. — I.  Form  each  couplet  of  the  compound  ratio 
from  the  numbers  given,  by  comparing  those  which  are  of 
the  same  unit  value,  arranging  the  terms  of  each  in  respect 
to  the  third  term  of  the  proportion,  as  if  it  were  the  first 
couplet  of  a  simple  proportion.    (767.) 

II.  Divide  the  product  of  the  second  and  third  terms  by 
the  product  of  the  first  terms,  the  quotient  will  be  the  num- 
ber required. 

The  same  preparation  of  the  terms  by  reduction  is  to  be  observed 
as  in  simple  proportion. 

When  possible,  shorten  the  operation  by  cancellation.  When 
the  vertical  line  is  used,  write  the  factors  of  the  dividend  on  the 
right,  and  the  factors  of  the  divisor  with  x  on  the  left. 


PROPORTION, 


397 


CAUSE     AND     EFFECT. 

773.  If  we  regard  the  conditions  of  each  problem  as 
the  comparison  of  two  causes  and  tivo  effects,  the  com- 
pound proportion  will  consist  of  two  ratios,  one  or  both 
of  which  may  be  compound,  and  the  required  term  will 
be  either  a  simple  cause,  or  effect,  or  a  single  element  of  a 
compound  cause,  or  effect. 


WRITTEN      EXERCISES. 

774.  1.  If  8  men  earn  $320  in  8  days,  how  much  will 
12  men  earn  in  4  days  ? 


1st  cause. 

8  men 
8  days 


$x 


STATEMENT. 
2d.  cause. 

4  days  ) 

OPERATION. 

12x4x^0  5 


Or, 


$% 


$x$ 


$240 


$x 


$240 


Analysis.  — 
In  this  example 
•*-"      the  first  cause  is 
4      8  men  at  work  8 
$$#0  ^  days,  the  second 
cause  is  12  men 
at  work  4  days  ; 
the  two  form  a 
compound  ratio. 
The  effect  of  the  first  cause  is  $320  earned,  the  effect  of  the  sec- 
ond cause  is  $x  earned,  and  is  the  required  term ;  the  two  effects 
form  a  simple  ratio. 

The  value  of  the  required  term  depends  upon  two  conditions  : 
1st,  the  number  of  men  at  work  ;  2d,  the  number  of  days  they  work. 
Consider  each  condition  separately,  and  arrange  the  terms  of  the 
same  unit  value  in  couplets,  and  make  a  statement  in  the  same  man- 
ner as  in  simple  proportion.  The  required  term  being  an  extreme, 
is  found  by  (763,  3). 

2.  If  it  cost  $41.25  to  pave  a  sidewalk  5  ft.  wide  and 
75  ft.  long,  what  will  it  cost  to  pave  a  similar  walk  8  ft. 
wide  and  566  ft.  long  ? 


398 


PROPORTION 


3.  How  many  days  will  21  men  require  to  dig  a  ditch 
80  ft.  long,  3  ft.  wide,  and  8  ft.  deep,  if  7  men  can  dig  a 
ditch  60  ft.  long,  8  ft.  wide,  and  6  ft.  deep,  in  12  days  ? 


7:21 
12:    x 


STATEMENT. 
60 


Or, 


» 


80 
3 

8 


OPERATION. 


8 


x  = 


2*X*0X*X*      -$■-"    - 


X 

% 

u 

It 

00 

$0* 

% 

$ 

0 

$ 

3 

8 

2|da. 


Analysis.— In  this  example  the  causes  and  the  effects  each  form 
a  compound  ratio.  The  required  term  is  an  element  of  the  second 
cause  and  a  mean.  Hence  divide  the  product  of  the  extremes  by 
the  product  of  the  given  means,  and  the  quotient  is  the  required 
factor  or  term,  2|  da.  (763,  2). 

4.  If  4  horses  consume  48  bushels  of  oats  in  12  days, 
how  many  bushels  will  20  horses  consume  in  8  weeks  ? 

Eule. — I.  Of  the  given  numbers,  select  those  which  con- 
stitute the  causes,  and  those  which  constitute  the  effects, 
and  arrange  them  in  couplets  as  in  simple  cause  and  effect, 
putting  x  in  the  place  of  the  required  term. 

II.  If  the  required  term,  x,  be  an  extreme,  divide  the 
product  of  the  means  by  the  product  of  the  given  extremes  ; 
if  x  be  a  mean,  divide  the  product  of  the  extremes  by  the 
product  of  the  given  means  ;  the  quotient  will  be  the  re- 
quired term. 

Solve  the  following  by  either  of  the  foregoing  methods  : 

5.  What  sum  of  money  will  produce  $300  in  8  mo.,  if 
$800  produce  $70  in  15  months  ? 


PROPORTION.  399 

6.  If  20  reams  of  paper  are  required  to  print  800  copies 
of  a  book  containing  230  pages  each,  40  lines  on  a  page, 
how  many  reams  are  required  to  print  3000  copies  of 
400  pages  each,  35  lines  on  a  page  ? 

7.  If  10  men  can  cut  46  cords  of  wood  in  18  da.,  work- 
ing 10  hr.  a  day,  how  many  cords  can  40  men  cut  in 
24  da.,  working  9  hr.  a  day  ? 

8.  What  is  the  cost  36£  yards  of  cloth  1J  yd.  wide, 
if  2£  yards  If  yd.  wide,  cost  $3.37£  ? 

9.  A  contractor  employs  45  men  to  complete  a  work 
in  3  months ;  what  additional  number  of  men  must  he 
employ,  to  complete  the  work  in  2£  months? 

10.  If  a  vat  16  ft.  long,  7  ft.  wide,  and  15  ft,  deep 
holds  384  barrels,  how  many  barrels  will  a  vat  17J  ft. 
long,  10£  ft.  wide,  and  13  ft.  deep  hold  ? 

11.  What  is  the  weight  of  a  block  of  granite  8  ft.  long, 

4  ft.  wide,  and  10  in.  thick,  if  a  similar  block  10  ft.  long, 

5  ft.  wide,  and  16  in.  thick,  weigh  5200  pounds  ? 

12.  If  it  cost  $15  to  carry  20  tons  1£  miles,  what  will 

it  cost  to  carry  400  tons  £  of  a  mile  ?  13^^ 

13.  If  it  take  13500  bricks  to  build  a  wall  200  ft.  long,  $ 
20  ft.  high,  and  16  in.  thick,  each  brick  being  8  in.  long, 

4  in.  wide,  and  2  in.  thick,  how  many  bricks  10  in.  long,  ^  ^ 

5  in.  wide,  3 J  in.  thick,  will  be  required  to  build  a  wall  <w 
600  ft.  long,  24  ft.  high,  and  20  ft.  thick  ? 

14.  What  will  15  hogsheads  of  molasses  cost,  if  28-J- 
gallons  cost  $7|  ? 

15.  At  6Jd.  for  1}  yards  of  cotton  cloth,  how  many 
yards  can  be  bought  for  £10  6s.  8d.  ? 

16.  If  $750  gain  $202.50  in  4  yr.  6  mo.,  what  sum  will 
gain  $155.52  in  1  yr.  6  mo.  ? 


400  PROPORTION. 

17.  In  what  time  can  60  men  do  a  piece  of  work  that 
15  men  can  do  in  20  days  ? 

18.  If  2£  yd.  of  cloth  6  quarters  wide  can  be  made  from 
1  lb.  12  oz.  of  wool,  how  many  yards  of  cloth  4  quarters 
wide  can  be  made  from  70  lb.  of  wool  ? 

19.  If  the  use  of  $300  for  1  yr.  8  mo.  is  worth  $30,  how 
long,  at  the  same  rate,  may  $210.25  be  retained  to  be 
worth  $42,891  ? 

20.  A  farmer  has  hay  worth  $1 8  a  ton,  and  a  merchant 
has  flour  worth  $10  a  barrel.  If  the  farmer  ask  $21  for 
his  hay,  what  should  the  merchant  ask  for  his  flour  ? 

21.  How  many  men  will  be  required  to  dig  a  cellar 
45  ft.  long,  34.6  ft.  wide,  and  12.3  ft.  deep,  in  12  da.  of 
8.2  hr.  each,  if  6  men  can  dig  a  similar  one  22.5  ft.  long, 
17.3  ft.  wide,  and  10.25  ft.  deep,  in  3  da.  of  10.25  hr.  each? 

22.  If  a  bin  8  ft.  long,  4£  ft.  wide,  and  2£  ft.  deep, 
hold  67£  bu.,  how  deep  must  another  bin  be  made,  that  is 
18  ft.  long  and  3|  ft.  wide,  to  hold  450  bu.  ? 

23.  What  will  120  lb.  of  coffee  cost,  if  10  lb.  of  sugar 
cost  $1.25,  and  16  lb.  of  sugar  are  worth  5  lb.  of  coffee  ? 

24.  Two  men  have  each  a  farm.  A's  farm  is  worth 
$48.75,  and  B's  $43-J-  ;  but  in  trading  A  values  his  at  $60 
an  acre.     What  value  should  B  put  upon  his  ? 

25.  If  6  men  in  4  mo.,  working  26  da.  for  a  month, 
and  12  hr.  a  day,  can  set  the  type  for  24  books  of  300  pp. 
each,  60  lines  to  the  page,  12  words  to  the  line,  and  an 
average  of  6  letters  to  the  word,  in  how  many  months  of 
24  da.  each,  and  10  hr.  a  day,  can  8  men  and  4  boys  set 
the  type  for  10  books  of  240  pp.  each,  52  lines  to  the 
page,  16  words  to  the  line,  and  8  letters  to  the  word,  2 
boys  doing  as  much  as  1  man  ? 


ORAL    E  XE  RCIS  ES. 

775.  1.  If  John  has  10  marbles,  William  15  marbles, 
and  Charles  25  marbles,  what  part  of  the  whole  has  each  ? 

2.  Two  men  bought  a  barrel  of  flour  for  $9,  the  first 
paying  $4  and  the  second  $5.  What  part  of  the  flour 
belongs  to  each  ? 

3.  Three  men  bought  108  sheep,  and  as  often  as  the 
first  paid  $3,  the  second  paid  $4,  and  the  third  $5.  How 
many  sheep  should  each  receive  ? 

4.  If  $45  be  divided  between  two  persons,  so  that  of 
every  $5,  one  receives  $2,  and  the  other  $3,  how  many 
dollars  does  each  receive  ? 

5.  Two  men  hired  a  pasture  for  $36 ;  one  put  in  2 
horses  for  3  weeks,  the  other  3  horses  for  4  weeks.  What 
should  each  pay  ? 

DEFINITIONS. 

776.  Partnership  is  the  association  of  two  or  more 
persons  under  a  certain  name,  for  the  transaction  of  busi- 
ness with  an  agreement  to  share  the  gains  and  losses. 

777.  A  Firm,  Company  or  House  is  any  par- 
ticular partnership  association. 

778.  The  Capital  is  the  money  or  property  invested 
by  the  partners,  called  also  Investment,  or  Joint- Stoch. 


402  PAKTNEKSHIP. 

779.  The  Resources  of  a  firm  are  the  amounts  due 
it,  together  with  the  property  of  all  kinds  belonging  to 
it ;  called  also  Assets,  or  Effects. 

780.  The  Liabilities  of  a  firm  are  its  debts. 

781.  The  Net  Capital  is  the  excess  of  resources 
over  liabilities. 

WRITTEN   JEXJSIiCISBS. 

782.  To  apportion  gains  or  losses  according  to 
capital  invested. 

1.  A  and  B  engage  in  trade  ;  A  furnishes  $400  capital, 
B  $600.     They  gain  $250  ;  what  is  the  profit  of  each  ? 
1st  operation.     {By  Fractions.) 
$400,  A.'s  investment  =  y$&  =  f  of  the  whole  capital. 
600,  B.'s  "  =  #fr  =  |     - 

$1000,  whole      " 

$250  x  f  =  $100,  A.'s  share  of  the  gain. 
$250  x  |  =  $150,  B.'s    " 

2d  operation.     {By  Proportion.) 
$1000  (whole  cap.)  :  $400  (A.'s  in  v.)  : :  $250  (whole  gain)  :  A.'s  share. 
$1000  (whole  cap.)  :  $600  (B.'s  inv.)  : :  $250  (whole  gain) :  B.'s  share. 

3d  operation.    {By  Percentage.) 

$250  gain  is  -ffifo  =  25%  of  the  whole  capital. 

$400  x  .25  =  $100,  A.'s  gain  ;    $600  x  ;25  =  $150,  B.'s  gain. 

Analysis.— {1st  Method.)  Since  $400,  A.'s  investment,  is  -f^, 
or  f,  of  the  whole  capital,  he  is  entitled  to  £  of  the  gain,  or  $100 ; 
and  B  is  entitled  to  §  of  the  gain,  or  $150. 

2d  Method.  The  ratio  of  $1000,  the  whole  capital,  to  $400,  A.'s 
investment,  is  equal  to  the  ratio  of  $250,  the  whole  gain,  to  A.'s 
share  of  the  gain.     Hence  the  proportions,  etc. 

3d  Method.  Since  the  gain  is  25%  of  the  whole  capital,  each 
partner  is  entitled  to  25  %  of  his  investment  as  his  share  of  the  gain. 

The  third  method  (by  dividend)  is  that  generally  adopted  by  joint- 
stock  companies  having  numerous  shareholders. 


PARTNERSHIP.  403 

2.  At  the  end  of  the  year,  Norton,  Smith  &  Co.  take 
an  account  of  stock,  and  find  the  amount  of  merchandise, 
as  per  inventory,  to  be  $8400  ;  cash  on  hand,  $4850 ;  due 
from  sundry  persons,  $5273.  Their  debts  are  found  to 
amount  to  $4223.  S.  Norton's  investment  in  the  busi- 
ness is  $5000  ;  R.  Smith's,  $4000 :  and  C.  Woodward's, 
$2000.  Make  a  statement  showing  the  resources,  lia- 
bilities, net  capital,  and  net  gain :  and  find  each  part 
ner's  share  of  the  gain. 

OPERATION. 

Resources. 

Mdse.  as  per  inventory, $8400 

Cash  on  hand, 4850 

Debts  due  the  firm, 5273 

•  $18533 

Liabilities. 

Debts  due  to  sundry  persons, 4223 

Net  capital,      ....  $14300 

Investments. 

S.  Norton, $5000 

R.  Smith, 4000 

C.  Woodward, 2000 

Total  investments $11000 

Net  gain, $3300 

S.  Norton's  fractional  part,  ■&%%  -  T5T  of  $3300  =  $1500. 
R.  Smith's         »  "      AVu°u  =  A  of  $3300  =  $1200, 

C.  Woodward's  "  "      fl&fc  =  T2T  of  $8300  =  $  600. 

Proof.— $1500  +  $1200  +  $600  =  $3300,  total  gain. 


404  PARTNERSHIP. 

Rule  1.  Find  wlmt  fractional  part  each  partner's  tn- 
vestment  is  of  the  ivhole  capital,  and  take  such  part  of  the 
whole  gain  or  loss  for  his  share  of  the  gain  or  loss.     Or, 

2.  State  by  proportion,  as  the  whole  capital  is  to  each 
'partner's  investment,  resptectively,  so  is  the  ivhole  gain  or 
loss  to  each  partner's  share  of  the  gain  or  loss.     Or, 

3.  Find  what  per  cent,  the  gain  or  loss  is  of  the  ivhole 
capital,  and  talce  that  per  cent,  of  each  partner's  invest- 
ment for  his  share  of  the  gain  or  loss,  respectively. 

3.  A  furnishes  $4000,  B,  $2700,  and  C,  $2300,  to  pur- 
chase a  house,  which  they  rent  for  $720.  What  is  each 
one's  share  of  the  rent  ? 

4.  Four  persons  rent  a  farm  of  230  A.  C4  P.  at  $7£  an 
acre.  A  puts  in  288  sheep,  B,  320  sheep,  C,  384  sheep, 
and  D,  648  sheep  ;  what  rent  ought  each  to  pay  ? 

5.  Prime  &  Co.  fail  in  business ;  their  liabilities 
amount  to  $22000  ;  their  available  resources  to  $8800. 
They  owe  A  $4275,  and  B  $2175.50 :  what  will  each  of 
these  creditors  receive  ?    a  ^2i  ^^-r-    ^   '  m,  1 

6.  Four  persons  engage  in  manufacturing,  and  invest 
jointly  $22500.  At  the  expiration  of  a  certain  time,  A's 
share  of  the  gain  is  $2000,  B's  $2800.75,  C's  $1685.25, 
and  D's  $1014.     How  much  capital  did  each  put  in  ? 

7.  An  estate  worth  $10927.60  is  divided  between  two 
heirs  so  that  one  receives  \  more  than  the  other.  What 
does  each  receive  ? 

8.  Three  persons  engage  in  the  lumber  trade  with  a 
joint  capital  of  $37680.  A  puts  in  $6  as  often  as  B  puts 
in  $10,  and  as  ,often  as  C  puts  in  $14.  Their  annual  gain 
is  equal  to  C's  stock.     What  is  each  partner's  gain  ? 


PARTNERSHIP.  405 

9.  Ames,  Lyon  &  Co.  close  business  in  the  following 
condition  :  notes  due  the  firm  to  the  amount  of  $24843.75, 
cash  in  hand,  $42375.80,  due  on  account,  $26500,  mer- 
chandise per  inventory,  $175840.  Notes  against  the 
firm,  $14058.75,  due  from  the  firm  on  account,  $12375.80. 
Ames  invested  $60000,  Lyon,  $40000,  and  Clark  $25000. 
Make  a  statement  showing  the  total  amount  of  resources, 
liabilities,  investments,  net  capital,  net  gain,  and  each 
partner's  share  of  the  gain. 

T83.  To  apportion  gains  or  losses  according"  to 
amount  of  capital  invested,  and  time  it  is  employed. 

1.  Three  partners,  A,  B,  and  C,  furnish  capital  as  fol- 
lows :  A,  $500  for  2  mo. ;  B,  $400  for  3  mo. ;  C,  $200  for 
4  mo.     They  gain  $600  ;  what  is  each  partner's  share  ? 

OPERATION. 

500  x  2  =  1000  =  flHHJ  =  i  x  )  (  $200,  A's  share. 

400  x  3  =  1200  =  ftflf  =  |  x  t  $600  =  J  $240,  B's      " 

200  x  4  =_800  =  380%°ff  =^sx  )  (  $160,  C's      " 

3000 

Analysis.— The  use  of  $500  for  2  mo.  is  the  same  as  the  use  of 
$1000  for  1  mo. ;  the  use  of  $400  for  3  mo.  is  the  same  as  that  of 
$1200  for  1  mo. ;  and  the  use  of  $200  for  4  mo.  is  the  same  as  that 
of  $800  for  1  mo.  Therefore  the  whole  capital  is  the  use  of  $3000 
for  1  mo.  ;  and  as  A's  investment  is  $1000  for  1  mo.,  it  is  £  of  the 
capital,  and  hence  he  should  receive  |  of  the  gain,  or  $200.  For 
the  same  reason,  B  should  receive  §,  and  C  T45  of  the  gain,  or  $240 
and  $160,  respectively. 

The  other  methods  of  operation  may  be  applied  by  considering 
the  products  of  investment  and  time  as  shares  of  the  capital.  Thus, 
$600  is  20%  of  $3000;  and  20%  of  $1000,  $1200,  and  $800  will 
give  $200,  $240,  and  $160,  respectively,  the  shares  of  gain  required 


406  PARTNERSHIP. 

2.  Barr,  Banks  &  Co.  gain  in  trade  $8000.  Barr  fur- 
nished $12000  for  0  mo.,  Banks,  $10000  for  8  mo.,  and 
Butts  $8000  for  11  mo.     Apportion  the  gain  ? 

Eule  1. — Multiply  each  partner's  capital  by  the  time 
it  is  invested,  and  divide  the  whole  gain  or  loss  among  the 
partners  in  the  ratio  of  these  products.     Or, 

2.  State  by  proportion :  TJie  sum  of  the  products  is  to 
each  product,  as  the  ivhole  gain  or  loss  is  to  each  partner's 
gain  or  loss. 

3.  Jan.  1,  1876,  three  persons  began  business  with 
$1300  capital  furnished  by  A  ;  March  1,  B  put  in  $1000  ; 
Aug.  1,  0  put  in  $900.  The  profits  at  the  end  of  the 
year  were  $750.     Apportion  it. 

4.  In  a  partnership  for  2  years,  A  furnished  at  first 
$2000,  and  10  mo.  after  withdrew  $400  for  4  mo.,  and 
then  returned  it ;  B  at  first  put  in  $3000,  and  at  the  end 
of  4  mo.  $500  more,  but  drew  out  $1500  at  the  end  of  16 
mo.     The  whole  gain  was  $3372.  Find  the  share  of  each. 

5.  The  joint  capital  of  a  company  was  $5400,  which 
was  doubled  at  the  end  of  the  year.  A  put  in  £  for  9  mo., 
B  f  for  6  mo.,  and  O  the  remainder  for  1  year.  What  is 
each  one's  share  of  the  stock  at  the  end  of  the  year  ? 

6.  Crane,  Child  &  Coe,  forming  a  partnership  Jan.  1, 
1875,  invested  and  drew  out  as  follows:  Crane  invested 
$2000,  4  mo.  after  $1000  more,  and  at  the  end  of  9  mo. 
drew  out  $600.  Child  invested  $5000,  6  mo.  after  $1200 
more,  and  at  the  end  of  11  mo.  put  in  $2000  more.  Coe 
put  in  $6000,  4  mo.  after  drew  out  $4000,  and  at  the 
end  of  8  mo.  drew  out  $1000  more.  The  net  profits  for 
the  year  were  $7570.     Find  the  share  of  each.     ,  - -tu  (H^ 

*/*    -J  8  DX'"    [    .     , 


784.  Alligation  treats  of  mixing  or  compounding 
two  or  more  ingredients  of  different  values  or  qualities. 

785.  Alligation  Ifedial  is  the  process  of  finding 
the  mean  or  average  value  or  quality  of  several  ingredients. 

786.  Alligation  Alternate  is  the  process  of  find- 
ing the  proportional  quantities  to  be  used  in  any  required 
mixture. 


WRITTEN     EXAMPLES 


787.  1.  If  a  grocer  mix  8  lb.  of  tea  worth  $.60  a  pound 
with  6  lb.  at  $.70,  2  lb.  at  $1.10,  and  4  lb.  at  $1.20,  what 
is  1  lb.  of  the  mixture  worth  ? 


OPERATION. 

$.60  x  8=    $4.80 

.70x6=      4.20 

1.10x2=      2.20 

1.20  x4=      4.80 

20  )  $16.00 


Analysis.  —Since  8  lb.  of  tea  at  $.60  is 
worth  $4.80,  and  6  lb.  at  $.70  is  worth 
$4.20,  and  2  lb.  at  $1.10  is  worth  $2.20, 
and  4  lb.  at  $1.20  is  worth  $4.80,  the  mix- 
ture of  20  lb.  is  worth  $16.  Hence  1  lb.  is 
worth  Jq  of  $16,  or  $16  -=-  20  =  $.80. 


2.  If  20  lb.  of  sugar  at  8  cents  be  mixed  with  24  lb.  at 
9  cents,  and  32  lb.  at  11  cents,  and  the  mixture  is  sold 
at  10  cents  a  pound,  what  is  the  gain  or  loss  on  the  whole  ? 

Rule.— Find  the  entire  cost  or  value  of  the  ingredients, 
and  divide  it  by  the  sum  of  the  simples. 

&/*7v    f   l 


408  ALLIGATION. 

3.  A  miller  mixes  18  bu.  of  wheat  at  $1.44  with  6  bu. 
at  $1.32,  6  bu.  at  $1.08,  and  12  bu.  at  $.84.  What  will  be 
his  gain  per  bushel  if  he  sell  the  mixture  at  $1.50  ? 

4.  Bought  24  cheeses,  each  weighing  25  lb.,  at  7?  a 
pound ;  10,  weighing  40  lb.  each,  at  lOf  ;  and  4,  weigh- 
ing 50  lb.  each,  at  13^ ;  sold  the  whole  at  an  average 
price  of  9J^  a  pound.     What  was  the  whole  gain  ? 

5.  A  drover  bought  84  sheep  at  $5  a  head  ;  96  at  $4.75  ; 
and  130  at  $5|-.  At  what  average  price  per  head  must 
he  sell  them  to  gain  20$  ? 

788.  To  find  the  proportional  parts  to  be  used, 
when  the  mean  price  of  a  mixture  and  the  prices  of 
the  simples  are  given. 

1.  What  relative  quantities  of  timothy  seed  worth  $2  a 
bushel,  and  clover  seed  worth  $7  a  bushel,  must  be  used 
to  form  a  mixture  worth  $5  a  bushel  ? 

operation.  Analysis. — Since  on  every  ingredient  used 

(2 


7 


2  )  whose  price  or  quality  is  less  than  the  mean 

Y  Ans.      rate  there  will  be  a  gain,  and  on  every  ingre- 
dient whose  price  or  quality  is  greater  than 


the  mean  rate  there  will  be  a  loss,  and  since  the  gains  and  losses 
must  be  exactly  equal,  the  relative  quantities  used  of  each  should 
be  such  as  represent  the  unit  of  value.  By  selling  one  bushel  oi 
timothy  seed  worth  $2,  for  $5,  there  is  a  gain  of  $3  ;  and  to  gain  $1 
would  require  *  of  a  bushel,  which  is  placed  opposite  the  2.  By 
selling  one  bushel  of  clover  seed  worth  $?,  for  $5,  there  is  a  loss 
of  $2  ;  and  to  lose  $1  would  require  ^  of  a  bushel,  which  is  placed 
opposite  the  7. 

In  every  case,  to  find  the  unit  of  value,  divide  $1  by  the  gain  or 
loss  per  bushel  or  pound,  etc.  Hence,  if  every  time  -*  of  a  bushel 
of  timothy  seed  is  taken,  £  of  a  bushel  of  clover  seed  is  taken,  the 
gain  and  loss  will  be  exactly  equal,  and  i  and  £  will  be  the  propor- 
tional  quantities  required. 


ALLIGATION 


409 


OPERATION. 


f 

1 

2 

3 

4 

5 

3 

i 

4 

4 

<     4 

i 

1 

1 

7 

1 

2 

2 

.10 

i 

3 

3 

To  express  the  proportional  numbers  in  integers,  reduce  these 
fractions  to  a  common  denominator,  and  use  their  numerators,  since 
fractions  having  a  common  denominator  are  to  each  other  as  their 
numerators  (241) ;  thus,  £  and  i  are  equaljo  f  and  f ,  and  the  pro- 
portional quantities  are  2  bu.  of  timothy  seed  to  3  bu.  of  clover  seed. 

2.  What  proportions  of  teas  worth  respectively  3,  4,  7, 
and  10  shillings  a  pound,  must  be  taken  to  forn^a  mix- 
ture worth  6  shillings  a  pound  ? 

Analysis. — To  preserve  the  equality 
of  gains  and  losses,  always  compare 
two  prices  or  simples,  one  greater  and 
one  less  than  the  mean  rate,  and  treat 
each  pair  or  couplet  as  a  separate  ex- 
ample. In  the  given  example  form  two 
couplets,  and  compare  either  3  and  10> 
4  and  7,  or  3  and  7,  4  and  10. 

We  find  that  ^  of  a  lb.  at  3s.  must  be 
taken  to  gain  1  shilling,  and  {of  a  lb. 
at  10s.  to  lose  1  shilling ;  also  \  of  a  lb.  at  4s.  to  gain  1  shilling,  and 
1  lb.  at  7s.  to  lose  1  shilling.  These  proportional  numbers,  obtained 
by  comparing  the  two  couplets,  are  placed  in  columns  1  and  2.  If, 
now,  the  numbers  in  columns  1  and  2  are  reduced  to  a  common  de- 
nominator, and  their  numerators  used,  the  integral  numbers  in 
columns  3  and  4  are  obtained,  which,  being  arranged  in  column  5, 
give  the  proportional  quantities  to  be  taken  of  each. 

It  will  be  seen  that  in  comparing  the  simples  of  any  couplet,  one 
of  which  is  greater,  and  the  other  less  than  the  mean  rate,  the  pro- 
portional number  finally  obtained  for  either  term  is  the  difference 
between  the  mean  rate  and  the  other  term.  Thus,  in  comparing  3 
and  10,  the  proportional  number  of  the  former  is  4,  which  is  the 
difference  between  10  and  the  mean  rate  6  ;  and  the  proportional 
number  of  the  latter  is  3,  which  is  the  difference  between  3  and  the 
mean  rate.  The  same  is  true  of  every  other  couplet.  Hence,  when 
the  simples  and  the  mean  rate  are  integers,  the  intermediate  steps 
taken  to  obtain  the  final  proportional  numbers  as  in  columns  1 ,  2,  3, 
and  4,  may  be  omitted,  and  the  same  results  readily  found  by  taking 
the  difference  between  each  simple  and  the  mean  rate,  and  placing 
it  opposite  the  one  with  which  it  is  compared. 


410  ALLIGATION. 

3.  In  what  proportions  must  sugars  worth  10  cents, 
11  cents,  and  14  cents  a  pound  be  used,  to  form  a  mix- 
ture worth  12  cents  a  pound  ? 

4.  A  farmer  has  sheep  worth  $4,  $5,  $6,  and  $8  per 
head.  What  number  may  he  sell  of  each  and  realize  an 
average  price  of  $5-|-  per  head  ? 

Rule. — I.  Write  the  several  prices  or  qualities  in  a 
column,  and  the  mean  price  or  quality  of  the  mixture  at 
the  left. 

II.  Form  couplets  by  comparing  any  price  or  quality 
less,  with  one  that  is  greater  than  the  mean  rate,  placing 
the  part  which  must  be  used  to  gain  1  of  the  mean  rale 
opposite  the  less  simple,  and  the  part  that  must  be  used  to 
lose  1  opposite  the  greater  simple,  and  do  the  same  for  each 
simple  in  every -couplet 

III.  If  the  proportional  numbers  are  fractional,  they 
may  be  reduced  to  integers,  and  if  two  or  more  stand  in 
the  same  horizontal  line,  they  must  be  added ;  the  final  re- 
sults tvill  be  the  proportional  quantities  required. 

1.  If  the  numbers  in  any  couplet  or  column  have  a  common  fac- 
tor, it  may  be  rejected. 

2.  We  may  also  multiply  the  numbers  in  any  couplet  or  column 
by  any  multiplier  we  choose,  without  affecting  the  equality  of  the 
gains  and  losses,  and  thus  obtain  an  indefinite  number  of  results, 
any  one  of  which  being  taken  will  give  a  correct  final  result. 

5.  What  amount  of  flour  worth  $5£,  $6,  and  $7J  per 
barrel,  must  be  sold  to  realize  an  average  price  of  $6£  per 
barrel  ? 

6.  In  what  proportions  can  wine  worth  $1.20,  $1.80, 
and  $2.30  per  gallon  be  mixed  with  water  so  as  to  form  a 
mixture  worth  $1.50  per  gallon  ? 


'30 

A 

4 

4 

24^ 

45 

tV 

8 

8 

48  ► 

.84 

A: 

A 

5 

5 

10 

GO. 

ALLIGATION.  411 

789.  When  the  quantity  of  one  of  the  simples  is 
limited. 

1.  A  farmer  has  oats  worth  $.30,  corn  worth  $.45,  and 

barley  worth  $.84  a  bushel.     To  make  a  mixture  worth 

$.60  a  bushel,  and  which  shall  contain  48  bu.  of  corn, 

how  many  bushels  of  oats  and  barley  must  he  use  ? 

operation.  Analysis.  —  By  the 

same  process  as  in 
(788),  the  proportional 
ns-  quantities  of  each  are 
found  to  be  4  bu.  of 
oats,  8  of  corn,  and  10 
of  barley.  But  since  48  bu.  of  corn  is  6  times  the  proportional  num- 
ber 8,  to  preserve  the  equality  of  gain  and  loss  take  G  times  the 
proportional  quantity  of  each  of  the  other  simples,  or  6  x  4  =  24  bu. 
of  oats,  and  6>  x  10  =  60  bu.  of  barley.     Hence,  etc. 

2.  A  dairyman  bought-  10  cows  at  $40  a  head.  How 
many  must  he  buy  at  $32,  $36,  and  $48  a  head,  so  that 
the  whole  may  average  $44  a  head  ? 

Kule. — Find  the  proportional  quantities  as  in  (788). 
Divide  the  given  quantity  by  the  proportional  quantity  of 
the  same  ingredient,  and  multiply  each  of  the  other  propor- 
tional quantities  by  the  quotient  thus  obtained. 

3.  A  grocer  having  teas  worth  $.80,  $1.20,  $1.50,  and 
$1.80  per  pound,  wishes  to  form  a  mixture  worth  $1.60  a 
pound,  and  use  20  lb.  of  that  worth  $1.50  a  pound. 

4.  Bought  12  yd.  of  cloth  for  $30.  How  many  yards 
must  I  buy  at  $3J  and  $1J  a  yard,  that  the  average  price 
of  the  whole  may  be  $2|  a  yard  ? 

5.  How  many  acres  of  land  worth  $70  an  acre  must  be 
added  to  a  farm  of  75  A.,  worth  $100  an  acre,  that  the 
average  value  may  be  $80  an  acre  ? 


412 


ALLIGATION 


10  ^ 


r 6 

i 

3 

3 

27 

7 

i 

2 

2 

18 

12 

i 

3 

3 

27 

,13 

t 

4 

4 

36 

12 

108 

790.  When  the  quantity  of  the  whole  compound 
is  limited. 

1.  A  grocer  has  sugars  worth  6  cents,  7  cents,  12  cents, 
and  13  cents  per  pound.  He  wishes  to  make  a  mixture 
of  108  pounds,  worth  10  cents  a  pound ;  how  many 
pounds  of  each  kind  must  be  use  ? 

operation.  Analysis. — The  proportion- 

al quantities  of  each  simple 
found  by  (788),  are  3  lb.  at 
C  cts.,  2  lb.  at  7  cts.,  3  lb.  at  12 
cts.,  and  4  lb.  at  13  cts.  Add- 
ing the  proportional  quantities, 
the  mixture  is  but  12  lb, 
while  the  required  mixture  is 
108,  or  9  times  12.  If  the 
whole  mixture  is  to  be  9  times  as  much  as  the  sum  of  the  propor- 
tional quantities,  then  the  quantity  of  each  simple  used  must  be  9 
times  as  much  as  its  respective  proportional,  or  27  lb.  at  6  cts., 
18  lb.  at  7  cts.,  27  lb.  at  12  cts..  and  36  lb.  at  13  cts. 

*  2.  A  man  paid  $330  per  week  to  55  laborers,  consisting 
of  men,  women,  and  boys;  to  the  men  he  paid  $10  a 
week,  to  the  women  $2  a  week,  and  to  the  boys  $1  a  week ; 
how  many  were  there  of  each  ? 

Rule. — Find  the  proportional  numbers  as  in  (788). 
Divide  the  given  quantity  by  the  sum  of  the  proportional 
quantities y  and  multiply  each  of  the  proportional  quanti- 
ties by  the  quotient  thus  obtained. 

3.  How  much  water  must  be  mixed  with  wine  worth 
$.90  a  gallon,  to  make  100  gal.  worth  $.  GO  a  gallon  ? 

4.  One  man  and  3  boys  received  $84  for  5G  days'  labor ; 
the  man  received  $3  per  day,  and  the  boys  $J,  $£,  and 
$lf  respectively  ;  how  many  days  did  each  labor  ? 


REVIEW. 


413 


791. 


RATIO. 


PROPOR- 
TION. 


SYNOPSIS  FOR  REVIEW. 

r  1.  Ratio.    2.  Sign  of  Ratio.    3.  Terms. 
4.  Antecedent.      5.   Consequent.      6. 

1.  Defs.  <       Value  of  a  Ratio.    7.  Simple  Ratio. 

8.  Compound  Ratio.    9.  Reciprocal  of 
(_      a  Ratio. 

2.  Formulas,  1,  2,  3. 

3.  General  Principles,  1,  2,  3. 

4.  General  Law. 


1.  Defs. 


1.  Proportion.      2.  Sign.       3.  Couplet. 
4.  Proportional.    5.  Antecedents.    6. 
Consequents.    7.  Extremes.   8.  Means. 
9.  Mean  Proportional. 
2.  Principles,  1,  2,  3,  4. 


1.  Defs 


3.  Simple  Pro- 
portion. 


4.  Compound 
Proportion. 


j  1 .  Simple  Proportion. 
'  \  2.  Statement. 

2.  Rule,  I,  II. 

3.  Cause  and  Effect. 

4.  Rule,  I,  II. 

f  1.  Def.    Compound  Proportion. 

2.  Rule,  I,  II. 

3.  Cause  and  Effect. 

4.  Rule,  I,  II. 


PARTNER- 
SHIP. 


ALLIGA- 
TION. 


{1.  Partnership.  2.  Firm,  Company,  or 
House.  3.  Capital.  4.  Resources. 
5.  Liabilities.      6.   Net   Capital. 

2.  782.    Rule,  1,  2,  3. 

3.  783.    Rule,  1,  2. 


1.  Defs 


)1.    Alligation. 
3. 


2.    Alligation  Medial. 
Alligation  Alternate. 

2.  787.    Rule, 

3.  788.    Rule,  I,  II,  III. 

4.  789.    Rule. 

5.  790.    Rule. 


414  GENERAL     REVIEW. 

TEST    PROBLEMS. 

792.  1.  The  sum  of  two  numbers  is  120,  and  their  dif- 
ference is  equal  to  £  of  the  greater.     Find  the  numbers. 

2.  E's  age  is  1J  times  the  age  of  D,  and  F's  age  is  2^ 
times  the  age  of  both,  and  the  sum  of  their  ages  is  124. 
What  is  the  age  of  each  ? 

3.  If  7  bu.  of  wheat  are  worth  10  bu.  of  rye,  and  5  bu. 
of  rye  are  worth  14  bu.  of  oats,  and  6  bu.  of  oats  are 
worth  $6,  how  many  bushels  of  wheat  will  $60  buy  ? 

4.  A  mechanic  was  engaged  to  labor  20  days,  on  the 
conditions  that  he  was  to  receive  $5  a  day  for  every  day 
he  worked,  and  to  forfeit  $2  a  day  for  every  day  he  was 
idle  ;  at  the  end  of  the  time  he  received  $86.  How  many 
days  did  he  labor  ? 

5.  One  man  can  build  a  fence  in  18  da.,  working  10  hr. 
a  day ;  another  can  build  it  in  9  da.,  working  8  hr.  a  day. 
In  how  many  days  can  both  together  build  it,  if  they 
work  6  hours  a  day  ? 

6.  If  6  boxes  of  starch  and  7  boxes  of  soap  cost  $33, 
and  12  boxes  of  starch  and  10  boxes  of  soap  cost  $54,  what 
is  the  price  of  1  box  of  each  ? 

7.  Three  men  agree  to  build  a  barn  for  $540.  The  first 
and  second  can  do  the  work  in  16  da.,  the  second  and 
third  in  13 I  da,,  and  the  first  and  third  in  llf  da.  In 
how  many  days  can  all  do  it  working  together  ?  In  how 
many  days  can  each  do  it  alone  ?  What  part  of  the  pay 
should  each  receive  ? 

8.  A  dealer  paid  $182  for  20  barrels  of  flour,  giving  $10 
for  first  quality,  and  $7  for  second  quality.  How  many 
barrels  were  there  of  each  ? 


GENERAL     REVIEW.  415 

9.  The  hour  and  minute  hands  of  a  clock  are  together 
at  12  if.  When  will  they  be  exactly  together  the  third 
time  after  this  ? 

10.  Bought  15  bu.  of  wheat  and  30  bu.  of  oats  for  $35, 
and  9  bu.  of  wheat  and  6  bu.  of  oats  for  $15.  What  was 
the  price  per  bushel  of  each  ? 

11.  If  Ames  can  do  as  much  work  in  3  days  as  Jones 
can  do  in  4J-  days,  and  Jones  can  do  as  much  in  9  days  as 
Smith  can  do  in  12  days,  and  Smith  as  much  in  10  days 
as  Ray  in  8  days,  how  many  days'  work  done  by  Kay  are 
equal  to  5  days  done  by  Ames  ? 

12.  A  merchant  bought  40  pieces  of  cloth,  each  piece 
containing  25  yd. ,  at  $4f  per  yard,  on  9  mo.  credit,  and 
sold  the  same  at  $4|  per  yard,  on  4  mo.  credit.  Find  his 
net  cash  gain,  money  being  worth  6%. 

13.  There  are  70  bu.  of  grain  in  2  bins,  and  in  1  bin 
are  10  bu.  less  than  f  as  much  as  there  is  in  the  other. 
How  many  bushels  in  the  larger  bin  ? 

14.  Three  men  can  perform  a  piece  of  work  in  12  hr. ; 
A  and  B  can  do  it  in  16  hr.,  A  and  C  in  18  hr.  What 
part  of  the  work  can  B  and  C  do  in  9-^-  hours  ? 

15.  What  per  cent,  in  advance  of  the  cost  must  a  mer- 
chant mark  his  goods,  so  that  after  allowing  5%  of  his 
sales  for  bad  debts,  an  average  credit  of  G  mo.,  and  4%  of 
the  cost  of  the  goods  for  his  expenses,  he  may  make  a 
clear  gain  of  Vl\%  on  the  first  cost  of  the  goods,  money 
being  worth  7%  ? 

16.  An  elder  brother's  fortune  is  1J-  times  his  younger 
brother's  ;  the  interest  of  £  of  the  elder  brother's  fortune 
and  £  of  the  younger's  for  5  years,  at  6$,  is  $2400.  What 
is  the  fortune  of  each  ? 


416  GENERAL     REVIEW. 

17.  The  top  of  Trinity  Church  steeple  in  New  York  is 
268  ft.  from  the  ground ;  J  the  height  of  the  steeple 
above  the  church  plus  12  ft.  is  equal  to  the  height  of  the 
church.    Find  the  height  of  the  steeple  above  the  church  ? 

1 8.  Two  persons  have  the  same  income  :  A  saves  J  of 
his,  but  B  by  spending  $300  a  year  more  than  A,  at  the 
end  of  2  years  is  $200  in  debt.     What  is  their  income  ? 

19.  Divide  $2520  among  3  persons,  so  that  the  second 
shall  have  f  as  much  as  the  first,  and  the  third  |  as  much 
as  the  other  two.     What  is  the  share  of  each  ? 

20.  A  man  owes  a  debt  to  be  paid  in  4  equal  install- 
ments of  4,  9,  12,  and  20  months  respectively  ;  a  discount 
of  5%  being  allowed,  he  finds  that  $1500  ready  money  will 
pay  the  debt.    What  is  the  amount  of  the  debt  ? 

21.  A  quantity  of  flour  is  to  be  distributed  among  some 
poor  families ;  if  50  lb.  are  given  to  each  family,  there 
will  be  6  lb.  left ;  if  51  lb.  are  given  to  each,  there  will  be 
wanting  4  lb.     What  is  the  quantity  of  flour  ? 

22.  1  have  three  notes  payable  as  follows  :  one  for  $400, 
due  Jan.  1,  1875  ;  another  for  $700,  due  Sept.  1,  1875 ; 
and  another  for  $1000,  due  April  1,  1876.  What  is  the 
average  of  maturity  ? 

23.  An  estate  worth  $123251.82  is  left  to  four  sons, 
whose  ages  are  19,  17,  13,  and  11  years,  respectively,  and 
is  to  be  so  divided  that  each  part  being  put  out  at  7% 
simple  interest,  the  amounts  shall  be  equal  when  they 
become  21  years  of  age.     What  are  the  parts  ? 

24.  If  a  piece  of  silk  cost  $1.20  a  yard,  at  what  price 
must  it  be  marked  that  it  may  be  sold  at  10%  less  than 
the  marked  price,  and  still  make  a  profit  of  20%  ? 

25.  A  farmer  sold  100  geese  and  turkeys  ;  for  the  geese 


GENERAL     REVIEW.  417 

he  received  $.75  apiece,  and  for  the  turkeys  $1.25  apiece, 
and  for  the  whole  $104.     What  was  the  number  of  each  ? 

26.  A  man  left  his  property  to  three  sons  ;  to  A  -J  want- 
ing $180,  to  B  £,  and  to  0  the  rest,  which  was  $590  less 
than  A  and  B  received.     What  was  the  whole  estate  ? 

27.  What  is  the  simple  interest  and  the  amount ;  the 
compound  interest  and  amount ;  the  present  worth  and 
the  true  discount ;  the  bank  discount  and  the  proceeds 
of  $1920,  for  2  yr.  5  mo.  12  da.,  at  6%  ?  Also  the  face  of 
the  note,  which  when  discounted  at  a  bank  for  the  same 
time,  and  at  the  same  rate,  will  produce  the  same  sum  ? 

28.  Divide  $1500  among  3  persons,  so  that  the  share 
of  the  second  may  be  £  greater  than  that  of  the  first,  and 
the  share  of  the  third  £  greater  than  that  of  the  second. 

29.  A  merchant  owes  for  three  bills  of  goods  as  follows : 
$500  due  March  1,  $800  due  June  1,  and  $600  due  Aug.  1. 
He  wishes  to  give  two  notes  for  the  amount,  one  for  $1000, 
payable  April  1  ;  what  must  be  the  face,  and  when  the 
maturity,  of  the  other  ? 

30.  A  man  in  New  York  purchased  a  draft  on  Chicago 
for  $10640,  drawn  at  60  da.,  $10283.56.  What  was  the 
course  of  exchange  ? 

31.  B.  B.  Northrop,  through'  his  broker,  invested  a 
certain  sum  in  U.  S.  6's,  5-20,  at  107|,  and  twice  as  much 
in  U.  S.  5's  of  '81,  at  98£,  brokerage  on  each,  \%.  His 
income  from  both  investments  is  $1674.  How  much  did 
he  invest  in  each  kind  of  stock  ? 

32.  A,  B,  and  C  are  under  a  joint  contract  to  furnish 
6000  bu.  of  corn,  at  $.48  a  bushel ;  A's  corn  is  worth  $.45, 
B's  $.51,  and  O's  $.54  ;  how  many  bushels  must  each  put 
into  the  mixture  that  the  contract  may  be  fulfilled  ? 


418  GENERAL     REVIEW. 

33.  A  cask  contains  42j  IT.  S.  gallons  of  wine,  worth 
$44  per  gallon.  How  much  less  will  it  cost  in  U.  S. 
money,  at  the  rate  of  £1  2s.  per  the  Imperial  gallon  ? 

34.  A  garden  400  ft.  long  and  300  ft.  wide  has  a  walk 
20  ft.  wide  laid  off  from  each  of  its  two  sides.  "What  is 
the  ratio  hetween  the  area  of  the  walk  and  the  area  of 
what  remains  ? 

35.  A  commission  merchant  in  Charleston  received  into 
his  store  on  May  1,  1875,  1000  bbl.  of  flour,  paying  as 
charges  on  the  same  day,  freight  $175.48,  cartage  $56.25, 
and  cooperage  $8.37.  He  sold  out  the  shipment  as  fol- 
lows: June  3,  200  bbl.  @  $6.25  ;  June  30,  350  bbl.  @ 
$6.50  :  July  29,  400  bbl.  @  $6.12|  ;  Aug.  6,  50  bbl.  @  $6. 
Required,  the  net  proceeds,  and  the  date  when  they  shall 
be  accredited  to  the  owner,  allowing  commission  at  3%%, 
and  storage  at  2  cents  per  week  per  bbl. 

36.  Three  men  engage  in  manufacturing.  L  puts  in 
$3840  for  6  mo.  ;  M,  a  sum  not  specified  for  12  mo. ;  and 
N,  $2560  for  a  time  not  specified.  L  received  $4800  for 
his  capital  and  profits  ;  M,  $9600  for  his  j  and  N,  $4160 
for  his.     Required,  M's  capital  and  Ws  time. 

37.  My  expenditures  in  building  a  house,  in  the  year 
1874,  were  as  follows  :  Jan.  16,  $536.78  ;  Feb.  20,  $425.36  ; 
March  4,  $259.25  ;  April  24,  $786.36.  At  the  last  date  I 
sold  the  house  for  exactly  what  it  cost,  interest  at  6  per 
cent,  on  the  money  expended  added,  and  took  the 
purchaser's  note  for  the  amount.  What  was  the  face  of 
the  note  ? 

38.  A  man  bought  a  farm  for  $6000,  and  agreed  to  pay 
principal  and  interest  in  3  equal  annual  installments. 
What  was  the  annual  payment,  interest  being  6$  ? 


ORAL      EXERCISES, 

793.  1.  What  is  the  product  of  3  used  twice  as  a 

factor  ? 

2.  What  is  the  product  of  3  used  3  times  as  factor  ? 

3.  What  is  the  product  of  4  used  3  times  as  a  factor  ? 

4.  What  is  the  result  of  using  5  twice  as  a  factor  ? 

5.  What  is  the  product  of  J  used  twice  as  a  factor  ? 

G.  What  is  the  result  of  using  J  twice  as  a  factor  ? 
f ,  three  times  as  a  factor  ? 

7.  What  number  will  be  produced  by  using  .3  twice  as 
a  factor?     .7,  twice?     .4,  three  times  ?    .05,  twice  ? 

8.  A  room  is  9  ft.  on  each  side  ;  how  many  square  feet 
in  the  floor  ? 

9.  A  cubical  block  of  stone  is  4  ft.  on  each  edge  ;  how 
many  cubic  feet  docs  it  contain  ? 

DEFINITIONS. 

794.  A  Poiver  of  a  number  is  the  product  of  factors, 
each  of  which  is  equal  to  that  number.  Thus,  27  is  the 
third  power  of  3,  since  27  =  3  x  3  x  3. 

795.  Involution  is  the  process  of  finding  any  power 
of  a  number. 


420 


INVOLUTION. 


796.  The  Base  or  Root  of  a  power  is  one  of  the 
equal  factors  of  the  power.  Thus,  27  is  the  third  power 
of  3,  and  3  is  the  base,  or  root,  of  that  power. 

797.  The  Exponent  of  a  power  is  a  number  placed 
at  the  right  of  the  base  and  a  little  above  it,  to  show  how 
many  times  it  is  used  as  a  factor  to  produce  the  power. 
It  also  denotes  the  degree  of  the  power.     Thus, 

31  or  3  =    3,  the  first     power  of  3. 

32  =  3  x  3  =9,  the  second  power  of  3. 

33  =  3  x  3  x  3  =27,  the  third    power  of  3. 
3^  =  3x3x3x3  =  81,  the  fourth  power  of  3. 


798.  The  Square  of  a  num- 
ber is  its  second  power,  so  called 
because  the  number  of  superficial 
units  in  a  square  is  equal  to  the 
second  power  of  the  njimber  of 
linear  units  in  one  of  its  sides. 


feL                S                   V                     \ 

!"h-»ii 

;  . 

1 

7i\ 

3»  = 


27 


799.  The  Cube  of  a  num- 
ber is  its  third  power,  so 
called  because  the  number  of 
units  of  volume  in  a  cube  is 
equal  to  the  third  power  of 
the  number  of  linear  units 
in  one  of  its  edges. 


800.  A  Perfect  JPoiver  is  a  number  which  can  be 
resolved  into  equal  factors.  Thus,  25  is  a  perfect  power 
of  the  second  degree,  and  27  is  a  perfect  power  of  the 


third  degree. 


INVOLUTION 


421 


Analysis. — Since  using 
any  number  three  times 
as  a  factor  produces  the 
third  power  of  that  num- 


801.  Peinciple. — The  sum  of  the  exponents  of  two 
powers  of  the  same  number  is  equal  to  the  exponent  of  the 
product  of  those  powers.  Thus,  22  x  23=25 ;  for  22=2  x  2, 
and  23=2  x  2  x  2  ;  hence  22  x  23=2  x  2  x  2  x  2  y  2=2G. 

WRITTEN     EXERCISES. 

802.  To  find  any  poAver  of  a  number. 

1.  Find  the  third  power  of  35. 

OPERATION. 

35  =  351 ;   35  x  35  =  352  =  1225 

1225  x  35  =  353  =s  42875 

ber  (797),  35  x  35  x  35  =  353  =  42875. 

2.  Find  the  square  of  37.     Of  42.     Of  56.    Of  75. 

3.  Find  the  cube  of  15.  '  Of  18.     Of  42.     Of  54. 

4.  What  is  the  value  of  632?  of  483?  of  324  ?  of  125? 
Rule. — Find  the  product  of  as  many  factors,  each 

equal  to  the  given  number,  as  there  are  units  in  the  expo- 
nent of  the  required  power. 

5.  What  is  the  third  power  of  £  ? 

#»*      a     a  4x4x4      43       64 

OPEBATION.-(4)3  =  txfxf  =  ^-^  =  -3  =  r-. 

Rule. — A  fraction  may  be  raised  to  any  power  by  in- 
volving each  of  its  terms  separately  to  the  required  power. 

6.  What  is  the  square  of  ^  ?    The  cube  of  £f  ? 

7.  Raise  -^  to  the  4th  power.     2£  to  the  5th  power. 
Find  the  required  power  of  the  following  : 


8. 

25.42. 

12. 

.03422. 

16. 

(182J)2. 

9. 

1063. 

13. 

.56. 

17. 

(4.07J)*. 

10. 

(44i)2. 

14. 

36.023. 

18. 

OA)5. 

11. 

Oi)4. 

15. 

.403163. 

19. 

.00638. 

422 


INVOLUTION. 


Find  the  value  of  each  of  the  following  expressions 


20.  4.63  x  253. 

21.  6.754-(7i)2. 

22.  *of(*)»x(3f)». 


23. 
24. 
25. 


86  -f-  .4096. 
2.53x(12})2. 
(7.5)3 -(1^)3. 


26.     (4s  x  56  x  123)  -+-  (42  x  104  x  32). 


FORMATION  OF  SQUARES  AND  CUBES  BY  THE  ANALYT 
ICAL  METHOD. 

803.  To  find  the  square  of  a  number  in  terms  of 
its  tens  and  units. 

1.  Find  the  square  of  27  in  terms  of  its  tens  and  units. 

Analysis.— The  product  of  20 
+  7  by  7  is  20  x  7  +  72,  and  the 
product  of  20+  7  by  20  is  20* +  (30 
x7);  hence  20s  +  (2  x  20  x  7)  +  72, 
which  is  the  sum  of  these  partial 
products,  is  the  square  of  20+7 
or  27. 


OPERATION. 


27 

27 


20  +  7 

20  +  7 


189  =  20x7  +  72 

540=        202+20x7 
729  =  202+(2x20x7)+72 


Principle. — The  square  of  a  number  consisting  of  tens 
and  units,  is  equal  to  the  sum  of  the  squares  of  the  tens 
and  units  increased  hy  tioice  their  product. 

Geometrical  Illustration. 

Let  ABCD  be  a  square,  each  side 
of  which  is  27  feet,  and  let  lines  be 
drawn  as  represented  in  the  figure.  It 
is  evident  that  the  square  ABCD  (272) 
is  equal  to  the  sum  of  two  squares,  one 
of  which  is  the  square  of  tens  (202),  the 
other  the  square  of  the  units  (72),  to- 
gether with  two  rectangles  each  of 
whose  areas  is  20  x  7. 


INVOLUTION".  423 


2.  What  is  the  square  of  37  ? 

302  =900  T\ 

2  x  30  x  7  =    420 

7a=     49 

37*  =  1369     (803,  Pkin.) 

3.  Find  the  square  of  42  in  terms  of  its  tens  and  units 
In  like  manner  find  the  square 


4.  Of  48. 

6.  Of    98. 

8.  Of  105. 

10.  Of  197. 

5.  Of  56. 

7.  Of  125. 

9.  Of  225. 

11.  Of  342. 

804.  To  find  the  cube  of  a  number  in  terms  of 
its  tens  and  units. 

1.  Find  the  cube  of  25  in  terms  of  its  tens  and  units. 


252  = 

25  = 

OPEKATION. 

202+ (2x20x5) +52 
20  +  5 

252x   5  = 

252 x 20  = 

203 

(202X5)  +  (2x20x52)  +  53 
+  (2x202x5)+       (20x52) 

253  =  203+ (3  x  202  x  5)  +  (3  x  20  x  52) +  53 

Analysis.  —The  square  of  25  is  202  +  (2  x  20  x  5)  +  52.  (803,  Pein.) 
Multiplying  this  by  20  +  5  gives  the  cube  of  25. 

2.  Find  the  cube  of  34  in  terms  of  its  tens  and  units. 

Principle. — The  cube  of  a  number  consisting  of  tens 
and  units  is  equal  to  the  cube  of  the  tens,  plus  three  times 
the  product  of  the  square  of  the  tens  by  the  units,  plus 
three  times  the  product  of  the  tens  by  the  square  of  the 
units,  plus  the  cube  of  the  units. 


424 


INVOLUTION 


Geometrical  Illustration. 


Fig. 


The  volume  of  the 
cube  marked  A,  Fig.  1, 
is  203  ;  the  volume  of 
each  of  the  rectangu- 
lar solids  marked  B  is 
20  x  20  x  5,  or  20*  x  5  ; 
the  volume  of  each  of 
the  rectangular  solids 
marked  C,  in  Fig.  2,  is 
20x5x5,  or  20 x52; 
and  the  volume  of  the 
small  cube  marked  D 
is  53.  It  is  evident, 
that  if  all  these  solids 
are  put  together  as 
represented  in  Fig.  3, 
a  cube  will  be  formed, 
each  edge  of  which 
is  25. 

3.  Find  the  cube 
of  46. 

OPERATION. 

403=  64000 

402x  6x3  =  28800 

40x62x3r=   4320 

63=     216 

463=  97336 

In  like  manner 
find  the  cube 

4.  Of  48. 

5.  Of  64. 

6.  Of  95. 

7.  Of  125. 


805.  1.  What  are  the  two  equal  factors  of  25  ?    36  ? 

2.  What  are  the  three  equal  factors  of  27  ?    64  ?    125  ? 

3.  What  are  the  four  equal  factors  of  16  ?     81  ?    256  ? 

4.  Of  what  is  81  the  2d  power  ?    The  4th  power  ? 

DEFINITIONS. 

806.  The  Square  Moot  of  a  number  is  one  of  the 
two  equal  factors  of  that  number ;  the  Cube  Hoot  is 
one  of  the  three  equal  factors  of  that  number,  etc. 

Thus,  3  is  the  square  root  of  9,  2  is  the  cube  root  of  8,  etc. 

807.  Evolution  is  the  process  of  finding  the  root 
of  any  power  of  a  number. 

808.  The  Radical  Sign  is  V-  When  prefixed  to 
a  number,  it  indicates  that  some  root  of  it  is  to  be  found. 

809.  The  Index  of  the  root  is  a  small  figure  placed 
above  the  radical  sign  to  denote  what  root  is  to  be  found. 
When  no  index  is  written,  the  index  2  is  understood. 

Thus,  yTOO  denotes  the  square  root  of  100 ;  ^125  denotes  the 
cube  root  of  125  ;  *J/256  denotes  the  fourth  root  of  256  ;  and  so  on. 

Evolution,  or  both  involution  and  evolution,  may  be  indicated  in 
the  same  expression  by  a  fractional  exponent,  the  numerator  de- 
noting the  required  power  of  the  given  number,  and  the  denomina- 
tor the  root  of  that  power  of  the  number.     Thus, 

9j  is  equivalent  to  /y/9  ;  64  V,  to  ^64  ;  and  8 1,  to  the  cube  root 
of  the  second  power  of  8,  equivalent  to  ^/8q,  etc. 


426  EVOLUTION. 

EVOLUTION    BY    FACTORING. 

WRITTEN    EXERCISES. 

810.  To  find  any  root  of  a  number  by  factoring. 

1.  Find  the  cube  root  of  1728. 

OPERATION. 
3)1728 

3^576 
I ! Analysis. — A  number  that  is  a  perfect  cube,  is 

3)192  composed  of  three  equal  factors,  and  one  of  them 

n\  n  a  is  the  cube  root  of  that  number. 

^Z_li  The  prime  factors  of  1728  are  3,  3,  3,  2,  2,  2, 

2)32  2,  2,  2  ;  hence  1728  =  (3  x  2  x  2)  x  (3  x  2  x  2)  x 

2MB  (3x2x2);  therefore  the  cube  root  of  1728  is 

'- (3  x  2  x  2),  or  12. 

2)_8 

2)4 

2 

Eule. — Resolve  the  given  number  into  its  prime  factors  ; 
then,  to  produce  the  square  root,  take  one  of  every  two  equal 
factors ;  to  produce  the  cube  root  take  one  of  every  three 
equal  factors  ;  and  so  on. 

2.  Find  the  square  root  of  64.  Of  256.  Of  576.  Of  6561. 

3.  Find  the  cube  root  of  729.  Of  2744.  Of  9261.  Of  3375. 

GENERAL    METHOD    OF    SQUARE    ROOT. 

811.  A  Perfect  Square  is  a  number  which  has 
an  exact  square  root. 

812.  Pki^ciples. — 1.  The  square  of  a  number  ex- 
pressed by  a  single  figure  contains  no  figure  of  a  higher 
order  than  tens. 

2.  The  square  of  tens  contains  no  significant  figure  of  a 
lower  order  than  hundreds,  nor  of  a  higher  order  than 
thousands. 


SQUARE     ROOT.  427 

3.   TJie  square  of  a  number  contains  twice  as  many 

figures  as  the  number,  or  twicj  as  many  less  one.     Thus, 

l2  =        1,  102  =  100, 

92  =       81,  1002  =       10000, 

992  =  9801,  10002  =  1000000. 

Hence, 

-  4.  If  any  perfect  square  be  separated  into  periods  of  two 

figures  each,  beginning  with  units'  place,  the  number  of 

periods  will  be  equal  to  the  number  of  figures  in  the  square 

root  of  that  number. 

If  the  number  of  figures  in  the  number  is  odd,  the  left-hand 
psriod  will  contain  only  one  figure. 

WRITTEN    EXERCISES. 

813.  To  find  the  square  root  of  a  number. 

1.  Find  the  square  root  of  4356. 

operation.  Analysis. — Since  4356  con- 

43,56  ( GO -f  6  =  66     sists    of    two    periods,    its 

nM 3 POO  square  root  will  consist  of 

two  figures  (812,  Prin.  4). 

120  -f  G  =  126  )  756  Since  56  cannot  be  a  part  of 

756  the  square  of  the  tens  (812, 

Prin.  2),  the  tens  of  the  root 

must  be  found  from  the  first  period  43. 

The  greatest  number  of  tens  whose  square  is  contained  in  4800 
is  6.  Subtracting  3600,  which  is  the  square  of  6  tens,  from  the 
given  number,  the  remainder  is  756.  This  remainder  is  composed 
of  twice  the  product  of  the  tens  by  the  units,  and  the  square  of  the 
units  (803,  Prin.).  But  the  product  of  tens  by  units  cannot 
be  of  a  lower  order  than  tens  ;  hence  the  last  figure  6  cannot  be  a 
part  of  twice  the  product  of  the  tens  by  the  unite ;  this  double 
product  must  therefore  be  found  in  the  part  750. 

Now,  if  we  double  the  tens  of  the  root  and  divide  750  by  the 
result,  the  quotient  6    will  be  the  units'  figure  of  the  root,  or  a 


428  EVOLUTION. 

figure  greater  than  the  units'  figure.  This  quotient  cannot  be  toe 
small,  for  the  part  750  is  at  least  equal  to  twice  the  product  of  the 
tens  by  the  units  ;  but  it  may  be  too  large,  for  the  part  750,  be- 
sides the  double  product  of  the  tens  by  the  units,  may  contain  tens 
arising  from  the  square  of  the  units.    (812,  Pexn.  1.)    Subtracting 

6  x  120  +  62  or  6  x  120  +  6  from  756,  nothing  remains.  Hence  66  is 
the  required  root. 

1.  In  this  example,  120  is  a  partial  or  trial  divisor,  and  126  is  a 
complete  divisor. 

2.  If  the  root  contains  more  than  two  figures,  it  may  be  found  by 
a  similar  process,  as  in  the  following  example,  where  it  will  be 
seen  that  the  partial  divisor  at  each  step  is  obtained  by  doubling 
that  part  of  the  root  already  found. 

2.  Find  the  square  root  of  186624. 

OPERATION. 

18,66,24(400  +  30  +  2=432 

16  00  00        _.  •       .  +,      .  _. 
The  ciphers  on  the  right 

400  X  2  +  30  =  830  )  2  66  24      are  usually  omitted  for  the 
2  49  00      ea^e  °*  brevity.     Thus, 

400  x  2  +  30  x  2  +  2=862 )  1724  18,66,24  (432 

1724  16__ 

83)266,  etc. 

3.  Find  the  square  root  of  7225. 

4.  What  is  the  square  root  of  58564. 

Rule. — I.  Separate  the  given  number  into  periods  of  two 
figures  each,  beginning  at  the  units'  place. 

II.  Find  the  greatest  number  whose  square  is  contained 
in  the  period  on  the  left ;  this  tuill  be  the  first  figure  in  the 
root.  Subtract  the  square  of  this  figure  from  the  period  on 
the  left,  and  to  the  remainder  annex  the  next  period  to  form 
a  dividend. 


SQUARE     ROOT 


429 


III.  Divide  this  dividend,  omitting  the  figure  on  the 
right,  by  double  the  part  of  the  root  already  found,  and 
annex  the  quotient  to  that  part,  and  also  to  the  divisor  ; 
then  multijrty  the  divisor  thus  completed  by  the  figure  of 
the  root  last  obtained,  and  subtract  the  product  from  the 
dividend. 

IV.  If  there  are  more  periods  to  be  brought  down,  con- 
tinue the  operation  in  the  same  manner  as  before. 

1.  If  a  cipher  occur  in  the  root,  annex  a  cipher  to  the  trial  divi- 
sor, and  another  period  to  the  dividend,  and  proceed  as  before. 

2.  If  there  is  a  remainder  after  the  root  of  the  last  period  is 
found,  annex  periods  of  ciphers  and  continue  the  root  to  as  many 
decimal  places  as  are  required. 

Find  the  square  root 


5.  Of  9604. 

7.  Of  11881. 

9.  Of  2050624. 

6.  Of  13225. 

8.  Of  994009. 

10.  Of  29855296. 

11.  Find  the  square  root  of  |fj. 

Operation.— y'S! 

-_Vioo_10 

—   ,— —  —  TT- 

V121 

Kule. — Tlie  square  root  of  a  fraction  may  be  found  by 
extracting  the  square  root  of  the  numerator  and  denomina- 
tor separately. 

Mixed  numbers  may  be  reduced  to  the  decimal  form  before  ex- 
tracting the  root ;  or,  if  the  denominator  of  the  fraction  is  a  perfect 
square,  to  an  improper  fraction. 

In  extracting  the  square  root  of  a  number  containing  a  decimal, 
begin  at  the  units'  place,  and  proceed  both  toward  the  left  and  the 
right  to  separate  into  periods,  then  proceed  as  in  the  extraction  of 
the  square  root  of  integers. 

Extract  the  square  root 


12.  Offlf. 
14-  OfTfHr 


15.  Of  .001225. 

16.  Of  196.1369. 

17.  Of  2.251521. 


18.  Of58.1406±. 

19.  Ofl7f. 

20.  Of  10795.21. 


430 


EVOLUTION 


21.  What  is  the  square  root  of  3486784401  ? 

22.  What  is  the  square  root  of  9.0000994009  ? 

23.  Find  the  value  of  32*  to  G  decimal  places. 

24.  Find  the  square  root  of  2f  to  4  decimal  places. 

25.  Find  the  square  root  of  -J  to  5  decimal  places. 

26.  Find  the  value  of  .125*  to  5  decimal  places. 

Find  the  second  member  of  the  following  equations . 

28.  (36^  xV.T5^ 


31. 


27.  AA1369  +  V/1296=? 

29.  2.83  -f-  V J17649  =  ? 

30.  Vf^-(^)*-22=? 

32.  V^6896  +  .3729  x  J  of  -v^256=  ? 

33.  (7.2  -  a/2770T)5  -r-  (|)2  =  ? 

34.  (a/81  -  16*)  x  (a/169  +  25*)  =  ? 

35.  V2642  X  4.41  +  (5.3361)*  -  (2.33  x 


i 


V32        \/92 


Geometrical  Explanation  of  Square  Root. 

814.  What  is  the  length  of  one  side  of  a  square  whose 
area  is  729  square  feet  ? 

Fig.  1.  Let  Fig.  1  represent  a  square  whose  area 

is  729  square  feet.     It  is  required  to  find  the 
length  of  one  side  of  this  square. 

Since  the  area  of  a  square  is  equal  to  the 
square  of  one  of  its  sides,  a  side  may  be 
found  by  extracting  the  square  root  of  the 
area. 

Since  729  consists  of  two  periods,  its  square 
root  will  consist  of  two  figures.     The  great- 
est number  of  tens  whose   square  is  con- 
tained in  700  is  2     Hence  the  length  of  the  side  of  the  square  is 
20  feet  plus  the  units'  figure  of  the  root. 


SQUARE     EOOT.  431 

Removing  the  square  whose  side  is  20  feet  and  whose  area  is  400 
square  feet,  there  remains  a  surface  whose 
area  is  329  square  feet  (Fig.  2).     This  re-  FlG-  2- 

mainder  consists  of  two  equal  rectangles,  ..        ||| 

each  of  which  is  20  feet  long,  and  a  square      pH 
whose  side  is  equal  to  the  width  of  each  20 

rectangle.     The  units'  figure  of  the  root  H 

is  equal   to  the   width  of  one  of  these     j  J^L^O      pK 
rectangles. 

The  area  of  a  rectangle  is  equal  to  the 

product  of  its  length  and  width  (462) ;     fc ^j 

hence,  if  the  area  be  divided  by  the  length, 

the  quotient  will  be  the  width.  Now,  since  the  two  rectangles 
contain  the  greater  portion  of  the  329  square  feet,  2  x  20  or  40, 
the  length  of  the  two  rectangles,  may  be  used  as  a  trial  divisor  to 
find  the  width.  Dividing  329  by  40,  the  quotient  is  8.  But  this 
quotient  is  too  large  for  the  width  of  the  rectangles,  for  if  8  feet  is 
the  width,  the  area  of  Fig.  2  will  be  40  x  8  +  82  or  384  square  feet. 
Taking  7  feet  for  the  width  of  the  rectangles,  the  area  of  Fig.  2  is 
40  x  7  +  72  or  329  square  feet.  Hence  20  +  7  or  27  feet  is  the  length 
of  a  side  of  the  square  whose  area  is  729  square  feet. 

PROBLEMS. 

815.  1.  A  square  field  contains  1016064  square  feet. 
What  is  the  length  of  each  side  ? 

2.  A  square  farm  contains  361  A.  Find  the  length  of 
one  side. 

3.  A  field  is  208  rd.  long  and  13  rd.  wide.  What  is  the 
length  of  the  side  of  a  square  containing  an  equal  area  ? 

4.  If  251  A.  65  P.  of  land  are  laid  out  in  the  form  of  a 
square,  what  will  be  the  length  of  each  of  its  sides  ? 

5.  A  circular  island  contains  21170.25  P.  of  land.  What 
is  the  length  of  the  side  of  a  square  field  of  equal  area  ? 

6.  If  it  cost  $312  to  enclose  a  field  216  rd.  long  and 
24  rd.  wide,  what  will  it  cost  to  enclose  a  square  field 
of  equal  area  with  the  same  kind  of  fence  ? 


432  EVOLUTION. 

CUBE    ROOT. 

816.  A  Perfect  Cube  is  a  number  which  has  an 
exact  cube  root. 

81.7.  The  Cube  Hoot  of  a  number  is  one  of  the  three 
equal  factors  of  that  number.  Thus,  the  cube  root  of  125 
is  5,  since  5x5x5  =  125. 

818.  Peikciples. — 1.  The  cube  of  a  number  expressed 
by  a  single  figure  contains  no  figure  of  a  higher  order  than 
hundreds. 

2.  The  cube  of  tens  contains  no  significant  figure  of  a 
lower  order  than  thousands,  or  of  a  higher  order  than 
hundred  thousands. 

3.  The  cube  of  a  number  contains  three  times  as  many 
figures  as  the  number,  or  three  times  as  many,  less  one  or 
two.     Thus, 

l3  =             1  103  at                       1,000 

33  =           27  1003  sa                 1,000,000 

93  ==         729  10003  =         1,000,000,000 

993  ="  907,299  100003  =  1,000,000,000,000 

4.  If  any  perfect  cube  be  separated  into  periods  of  three 
figures  each,  beginning  with  units'  place,  the  number  of 
periods  will  be  equal  to  the  number  of  figures  in  the  cube 
root  of  that  number. 

WRITTEN     EXERCISES. 

819.  To  find  the  cube  root  of  a  number. 
1.  Find  the  cube  root  of  405224. 

OPERATION. 

405,224  (  70  +  4  =  74,  cube  root 
703  =  343  000 


702  x  3  =  14700  )  62  224 
743  ss  405  224 


CUBE     ROOT.  433 

Analysis. — Since  405224  consists  of  two  periods,  its  cube  root 
will  consist  of  two  figures  (818,  Purr.  4)  Since  224  cannot  be  a 
part  of  the  cube  of  the  tens  of  the  root  (818,  Prin.  2),  the  first 
figure  of  the  root  must  be  found  from  the  first  period,  405.  The 
greatest  number  of  tens  whose  cube  is  contained  in  405000  is  7. 
Subtracting  the  cube  of  7  tens  from  the  given  number,  the  remain- 
der is  62224  This  remainder  is  equal  to  the  product  of  three  times 
the  square  of  the  tens  of  the  root  by  the  units,  plus  three  times  the 
product  of  the  tens  by  the  square  of  the  units,  plus  the  cube  of  the 
units  (804,  Prln.)  But  the  product  of  the  square  of  tens  by  units 
cannot  be  of  a  lower  order  than  hundreds  (818,  Prin.  2);  hence 
the  number  represented  by  the  last  two  figures,  24,  cannot  be  a  part 
of  three  times  the  product  of  the  square  of  the  tens  of  the  root  by 
the  units  ;  the  triple  product  must  therefore  be  found  in  the  part 
62200.  Hence,  if  62200  be  divided  by  3  x  70*,  the  quotient,  which 
is  4,  will  be  the  units'  figure  of  the  root  or  a  figure  greater  than  the 
units'  figure.  Subtracting  743  from  the  given  number,  the  result 
is  0  ;  hence  74  is  the  required  root. 

Instead  of  cubing  74,  the  parts  which  make  up  the  remainder 
62224  may  be  formed  and  added  thus  : 


8 

X 

702 

x  4  = 

58800 

3 

x 

70 

x  42  = 
43  = 

3360 
64 

62224 ; 

Or,  since  4  is  a  common  factor  in  the  three  parts  which  make  up 
the  remainder,  these  parts  may  be  combined  thus 

3  x  702         =r  14700 
3  x  70  x  4  =     840 

42= 16 

15556x4=62224. 

1.  In  this  example,  14700  is  a  partial  or  trial  divisor,  and  15556  is 
a  complete  divisor. 

2.  If  the  cube  root  contains  more  than  two  figures,  it  may  be 
found  by  a  similar  process,  as  in  the  following  example,  where  it 
will  be  seen  that  the  partial  divisor  at  each  step  is  equal  to  three 
times  the  square  of  that  part  of  the  root  already  found. 


434 


EVOLUTION 


2.  Find  the  cube  root  of  12812904. 

OPERATION. 


2003= 


Cube  Root, 
12,812,904  ( 200  +  30  +  4=234 
8  000  000 


1  ST  PAR.  DIVISOR  3  X  200*        =  120000 

4  812  904 

3x200x30=  18000 

30*=      900 

41G7000 

1st  complete  divisor             138900 

645904 

2d  par.  divisor    3  x  2302     =158700 

3  x  230  x  4=     2760 

42=        10 

645904 

2d  complete  divisor              161476 

8 


The  operation  may  be  abridged  as  follows  : 

12,812,904(234 

23= 

1st  partial  divisor  3  x  202     =1200 

3x20x3=  180 

3»=      9 


1st  complete  divisor 


1389 


2d  par.  divisor 


2302     =158700 

230x4=     2760 

42=        16 


4812 


4167 


645904 


645904 


2d  complete  divisor  161476 

.Rule. — I.  Separate  the  given  number  into  periods  of 
three  figures  each,  begimiing  at  the  units'  place. 

II.  Find  the  greatest  number  whose  cube  is  contained  in 
the  period  on  the  left ;  this  will  be  the  first  figure  in  the 
root.  Subtract  the  cube  of  this  figure  from  the  period  on 
the  left,  and  to  the  remainder  annex  the  next  period  to 
form  a  dividend. 

III.  Divide  this  dividend  by  the  partial  divisor,  which 
is  3  times  the  square  of  the  root  already  found,  considered 
as  tens  ;  the  quotient  is  the  second  figure  of  the  root. 


CUBE     ROOT.  435 

IV.  To  the  partial  divisor  add  3  times  the  product  of  the 
second  figure  of  the  root  by  the  first  considered  as  tens,  also 
the  square  of  the  second  figure,  the  result  will  he  the  com- 
plete divisor. 

V.  Multiply  the  complete  divisor  by  the  second  figure  of 
the  root  and  subtract  the  product  from  the  dividend. 

VI.  //  there  are  more  periods  to  be  brought  down,  pro- 
ceed as  before,  using  the  part  of  the  root  already  found, 
the  same  as  the  first  figure  in  the  previous  process. 

1.  If  a  cipher  occur  in  the  root,  annex  two  ciphers  to  the  trial 
divisor,  and  another  period  to  the  dividend  ;  then  proceed  as  before, 
annexing  both  cipher  and  trial  figure  to  the  root. 

2.  If  there  is  a  remainder  after  the  root  of  the  last  period  is  found, 
annex  periods  of  ciphers  and  proceed  as  before.  The  figures  of  the 
root  thus  obtained  will  be  decimals. 

What  is  the  cube  root 


3.  Of  15625  ? 

4.  Of  166375  ? 


5.  Of  1030301 ? 

6.  Of  4492125  ? 


7.  Of  1045678375  ? 

8.  Of  4080659192  ? 


9.  Find  the  cube  root  of  ^-. 

3/- 

Operation.— ty£T  = 


Rule. — The  cube  root  of  a  fraction  may  be  found  by 
extracting  the  cube  root  of  the  numerator  and  denominator. 

In  extracting  the  cube  root  of  decimal  numbers,  begin  at  the 
units'  place  and  proceed  both  toward  the  left  and  the  right,  to 
separate  into  periods  of  three  figures  each. 

Extract  the  cube  root 


12.  Of  2f  I  14.  Of  .091125. 

13.  Of  39304.        15.   Of  12.812904. 


10.  Of  \^\. 

11-  Of  IfHf 

16.  What  is  the  cube  root  of  98867482624  ? 

17.  What  is  the  cube  root  of  .000529475129  ? 

18.  Find  the  cube  root  of  \  correct  to  4  decimal  places. 


436 


EVOLUTION. 


Find  the  second  member  of  the  following  equations 


19.  1.44*+  2.5*  =  ?     I     21. 

20.  V/Sf|x^Jfi=?|     22. 
23.  24.8  +  ^103.823  x  (.125)  a 
24. 


^.4096  —  .2368  =  ? 
V  54^872- (21.952)3=? 
? 


166  ■*■  #64  —  (4  x  #.512)  =  ? 

Geometrical  Explanation  of  Cube  Koot. 

820.  What  is  the  length  of  the  edge  of  a  cube  whose 
volume  is  15625  cubic  feet  ? 

^IG-  !■  Let  Fig.  1  represent  a 

cube  whose  volume  is 
15625  cubic  feet.  It  is 
required  to  find  the  length 
of  the  edge  of  this  cube. 

Since  the  volume  of  a 
cube  is  equal  to  the  cube 
of  one  of  its  edges,  an 
edge  may  be  found  by 
extracting  the  cube  root 
of  the  volume. 

Since  15625  consists  of 
two  periods,  its  cube  root 
will  consist  of  two  figures.  The  greatest  number  of  tens  whose 
cube  is  contained  in  15000  is  2.  Hence,  the  length  of  the  edge  of 
the  cube  is  20  feet  plus  the  units'  figure  of  the  root.  Removing  the 
cube  whose  edge  is  20  feet  and  whose  volume  is  8000  cubic  feet, 
there  remains  a  solid  whose  volume  is  7625  cubic  feet  (Fig.  2). 
This  remainder  consists  of  solids  similar  to  those  marked  B,  C,  and 
D,  in  Fig.  1  and  Fig.  2  of  Art.  804. 

15,625(25 
23=     8 
3 
3 


202    =  1200 

7625 

20x5=  300 

52=  25 

1525 

7625 

CUBE     ROOT. 


43? 


The  volume  of  a  rectan-  Fig.  2. 

gular  solid  is  equal  to  the 
product  of  the  area  of  its 
base  by  its  height  or  thick- 
ness (472) ;  hence,  if  the 
volume  be  divided  by  the 
area  of  the  base  the  quo- 
tient will  be  the  thickness. 
^ow,  since  the  three  equal 
rectangular  solids,  each 
of  which  is  20  feet  square 
and  whose  thickness  is  the 
units'  figure  of  the  root, 
contain  the  greater  por- 
tion of  the  7025  cubic  feet,  3  x  202  or  300  x  22  may  be  used  as  a  trial 
divisor  to  find  the  thickness.  Dividing  7625  by  1200  the  quotient 
is  6.  But  this  quotient  is  too  large,  for  if  6  feet  is  the  thickness, 
the  volume  of  Fig.  2  will  be  3x  202  x  6  +  3  x20x62  +  63,  or  957G 
cubic  feet.  Taking  5  feet  for  the  thickness,  the  volume  of  Fig.  2 
is  7625  cubic  feet,  for  3x202  x5  +  3  x20x52  +  53=(300x22  +  30x2 
x  5  +  5-)  5  =  1525  x  5=7625.  Hence,  25  feet  is  the  length  of  the  edge 
of  a  cube  whose  volume  is  15625  cubic  feet. 


PHO&ZEJXS. 

821.  1.  What  is  the  length  of  the  edge  of  a  cubical 
box  that  contains  46656  cu.  inches  ? 

2.  What  must  be  the  length  of  the  edge  of  a  cubical 
bin  that  shall  contain  the  same  volume  as  one  that  is 
16  ft.  long,  8  ft.  wide,  and  4  ft.  deep? 

3.  What  are  the  dimensions  of  a  cube  that  has  the 
same  volume  as  a  box  2  ft.  8  in.  long,  2  ft.  3  in.  wide,  and 
1  ft.  4  in.  deep  ? 

4.  How  many  square  feet  in  the  surface  of  a  cube 
whose  volume  is  91125  cubic  feet  ? 

5.  What  is  the  length  of  the  inner  edge  of  a  cubical 
bin  that  contains  150  bushels  ? 


438  EVOLUTION. 

6.  What  is  the  depth  of  a  cubical  cistern  that  holds 
200  barrels  of  water  ? 

7.  Find  the  length  of  a  cubical  vessel  that  will  hold 
4000  gallons  of  water. 

ROOTS  OF  HIGHER  DEGREE. 

822.  Any  root  whose  index  contains  no  other  factors 

than  2  or  3  may  be  extracted  by  means  of  the  square  and 

cube  roots. 

If  any  power  of  a  given  number  is  raised  to  any  required  power, 
the  result  is  that  power  of  the  given  number  denoted  by  the  pro- 
duct of  the  two  exponents.  (801.)  Conversely,  if  two  or  more 
roots  of  a  given  number  are  extracted,  successively,  the  result  is 
that  root  of  the  given  number  denoted  by  the  product  of  the  indices. 

1.  What  is  the  6th  root  of  2176782330  ? 

operation.  Analysis. — The  index  of  the  re- 

^2176782336  =  46656         <*uired  root  «•  6  « t  x  •  j  hence  ex- 

tract  the  square  root  of  the  given 

V  46656  =36  number,  and  the  cube  root  of  this 

Or  result,  which  gives  30  as  the  6th  or 

*8/917ft7«9qqn  —  19Qfi  required  root.     Or,   first   find  the 

V 217678233b  _  129b  cube  poot  of  the  giyen  numberj  and 

Y  1296  =  36  then  the  square  root  of  the  result. 

Rule. — Separate  the  index  of  the  required  root  into  its 
prime  factors,  and  extract  successively  the  roots  indicated 
by  the  several  factors  obtained  ;  the  final  result  will  be  the 
required  root. 

2.  What  is  the  4th  root  of  5636405776  ? 

3.  What  is  the  8th  root  of  1099511627776  ? 

4.  What  is  the  6th  root  of  25632972850442049? 

5.  What  is  the  9th  root  of  1.577635  ? 

For  further  practical  applications  of  Involution  and  Evolution, 
see  "  Mensuration." 


823.  An  Arithmetical  Progression  is  a  suc- 
cession of  numbers,  each  of  which  is  greater  or  less  than 
the  preceding  one  by  a  constant  difference. 

Thus,  5,  7,  9,  11,  13,  15,  is  an  arithmetical  progression. 

824.  The  Terms  of  an  arithmetical  progression  are 
the  numbers  of  which  it  consists.  The  first  and  last  terms 
are  called  the  Extremes,  and  the  other  terms  the  Means. 

825.  The  Common  Difference  is  the  difference 
between  any  two  consecutive  terms  of  the  progression. 

826.  An  Increasing  Arithmetical  Progres- 
sion is  one  in  which  each  term  is  greater  than  the  pre- 
ceding one. 

Thus,  1,  3,  5,  7,  9,  11,  is  an  increasing  progression. 

827.  A  Decreasing  Arithmetical  Progres- 
sion is  one  in  which  each  term  is  less  than  the  preced- 
ing one. 

Thus,  15,  13,  11,  9,  7,  5,  3,  1,  is  a  decreasing  progression. 

828.  The  following  are  the  quantities  considered  in, 
arithmetical  progression  and  the  abbreviations  used  for 
them : 


1.  The  first  term,  (a). 

2.  The  last  term,  (/). 


3.  The  common  difference,  (d). 

4.  The  number  of  terms,      (n). 


5.  The  sum  of  all  the  terms,  (s). 


440  PROGRESSIONS. 

WRITTEN      EXERCISES. 

829.  To  find  one  of  the  extremes,  when  the  other 
extreme,  the  common  difference,  and  the  number 
of  terms  are  given. 

1.  The  first  term  of  an  increasing  progression  is  8,  the 
common  difference  5,  and  the  number  of  terms  20 ;  what 
is  the  last  term  ? 

operation.  Analysis.— The  2d  term  is  8  +  5; 

20  —  1  =  19  the  3d  term  is  8  +  (5  x  2)  the  4th  term 

Yq k    ,    o  -j  nq 7  is  8  +  (5  x  3) ;  and  so  on.     Hence  8  + 

UXO  +  »-lW-i.  (19  x  5)  or  103  is  the  20th  or  last  term. 

2.  The  last  term  of  an  increasing  progression  is  103, 

the  common  difference  5,  and  the  number  of  terms  20 ; 

what  is  the  first  term  ? 

operation  Analysis. — The  1st  term  must  be 

0         1  1  Q  a  number  to  which,  if  19  x  5  be  added, 

zi)  —  i  _^iy^  the  gum  gliall  be  103 .  hencGj  if  19  x  5 

103  —  19x5  =  8  =  $     is  subtracted  from  103,  the  remainder 
is  the  first  term. 

3.  The  first  term  of  a  decreasing  progression  is  203, 
the  common  difference  5,  and  the  number  of  terms  40 ; 
what  is  the  last  term  ? 

4.  The  last  term  of  a  decreasing  progression  is  1,  the 
common  difference  2,  and  the  number  of  terms  9  ;  what 
is  the  first  term  ? 

Rule.— I.  If  the  given  extreme  is  the  less,  add  to  it  the 
product  of  the  common  difference  by  the  number  of  terms 
less  one. 

II.  If  the  given  extreme  is  the  greater,  subtract  from  it 
the  product  of  the  common  difference  by  the  number  of 

terms  less  one. 

_  U=a+(«-l)xrf. 

FORMULiE.—  i  7         ;  '         , 

'  a  =  I  —  (w  —  1)  x  a. 


PROGRESSIONS.  441 

5.  The  first  term  of  an  increasing  progression  is  5,  the 
common  difference  4,  and  the  number  of  terms  8  ;  what 
is  the  last  term  ? 

6.  The  first  term  of  an  increasing  progression  is  2,  and 
the  common  difference  3  ;  what  is  the  50th  term  ? 

7.  The  first  term  of  a  decreasing  progression  is  100, 
and  the  common  difference  7 ;  what  is  the  13th  term  ? 

8.  The  first  term  of  an  increasing  progression  is  f,  the 
common  difference  f ,  and  the  number  of  terms  20  ;  what 
is  the  last  term  ? 

830.  To  find  the  common  difference,  when  the 
extremes  and  number  of  terms  are  given. 

1.  The  extremes  of  a  progression  are  8  and  103,  and 
the  number  of  terms  20  ;  what  is  the  common  difference  ? 

operation.  Analysis.— The  difference  between 

ino        q  _i_  -i q  __  k  __  j       the  extremes  is  equal  to  the  product 

of  the  common  difference  by  the 
number  of  terms  less  one  (829) ;  hence  the  common  difference  is 
Hi  or  5. 

2.  The  extremes  of  a  progression  are  1  and  17,  and  the 
number  of  terms  9  ;  what  is  the  common  difference  ? 

Eule. — Divide  the  difference  between  the  extremes  by 
the  number  of  terms  less  one. 

Formula. — d  =  — — - . 
n  —  1 

3.  The  extremes  are  3  and  15,  and  the  number  of  terms 
7  ;  what  is  the  common  difference  ? 

4.  The  extremes  are  1  and  51,  and  the  number  of  terms 
76  ;  what  is  the  common  difference  ? 


442  PROGRESSIONS. 

5.  The  youngest  of  ten  children  is  8,  and  the  eldest  44 
years  old ;  their  ages  are  in  arithmetical  progression. 
What  is  the  common  difference  of  their  ages  ? 

6.  The  amount  of  $800  for  60  years,  at  simple  interest, 
is  $4160.     What  is  the  rate  per  cent.  ? 

7.  The- extremes  are  0  and  2|,  and  the  number  of  terms 
18  ;  what  is  the  common  difference  ? 

831.  To  find  the  number  of  terms,  when  the  ex- 
tremes and  common  difference  are  given. 

1.  The  extremes  of  a  progression  are  8  and  103,  and 
the  common  difference  5  ;  what  is  the  number  of  terms  ? 

operation.  Analysis. — The  difference  between  the 

iaq        6  m£m  k  _  29        extremes  is  equal  to  the  product  of  the 

*  common    difference    by    the   number   of 

iy-f-l        /CO        ill       terms  less  one  (830) ;  hence  the  number 

of  terms  less  one  is  equal  to  -9/  or  19 ; 

therefore  19  + 1  or  20  is  the  number  of  terms. 

2.  The  extremes  of  a  progression  are  1  and  17,  and  the 
common  difference  2  ;  what  is  the  number  of  terms  ? 

Rule. — Divide  the  difference  between  the  extremes  by 
the  common  difference,  and  add  one  to  the  quotient. 

Formula. — n  =  -^ — h  1« 
d 

3.  The  extremes  are  5  and  75,  and  the  common  differ- 
ence is  5  ;  what  is  the  number  of  terms  ? 

4.  The  extremes  are  J  and  20,  and  the  common  differ- 
ence is  6J- ;  what  is  the  number  of  terms  ? 

5.  A  laborer  received  50  cents  the  first  day,  54  cents 
the  second,  58  cents  the  third,  and  so  on,  until  his  wages 
were  $1.54  a  day  ;  how  many  days  did  he  work  ? 

6.  In  what  time  will  $500,  at  7  per  cent,  simple  inter- 
est, amount  to  $885  ? 


PROGRESSIONS.  443 

832.  To  find  the  sum  of  all  the  terms,  when  the 
extremes  and  the  number  of  terms  are  given. 

1.  The  extremes  of  an  arithmetical  progression  are  2 
and  14,  and  the  number  of  terms  is  5  ;  what  is  the  sum 
of  all  the  terms  ? 

Analysis. — The  common  dif- 

operation.  erence  is  found  to  be  3  (830) ; 

2_i_    5_i_    84-114-14       hence    the    required    sum   is 


s  =14  +  11+  8+   5+   2 


equal  to  2-1-5  +  8  +  11  +  14,  or 
14  +  11  +  8  +  5  +  2.    Adding  the 


2  S  =16  +  16  +  16  +  16  +  16  corresponding  terms  of  these 

2  S  =  16  X  5  =  (2  +  14)  X  5  two  progressions,  we  have  2 

g    ,    i  a  •  times  the  sum  =  16  x  5  =  (2  + 

X  5  =  40.  14)  x  5  ;    hence    the    sum    is 

2  +  14      r        An 

— - —  x  5  =  40. 


2 


2.  The  extremes  of  an  arithmetical  progression  arc  5 
and  75,  and  the  number  of  terms  is  15  ;  what  is  the  sum. 
of  all  the  terms  ? 

Rule. — Multiply  the  sum  of  the  extremes  by  half  the 
number  of  terms. 

Formula. — s  =  -  x  («  +  ?)• 

3.  The  extremes  are  4  and  40,  and  the  number  of  terms 
is  7  ;  what  is  the  sum  of  all  the  terms  ? 

4.  The  extremes  are  0  and  250,  and  the  number  of 
terms  is  1000  ;  what  is  the  sum  of  all  the  terms  ? 

5.  How  many  strokes,  beginning  at  1  o'clock,  does  the 
hammer  of  a  common  clock  strike  in  12  hours  ? 

6.  A  body  will  fall  16^  ft.  in  the  first  second  of  its 
fall,  48J  ft.  in  the  second  second,  80T5F  ft.  in  the  third 
second,  and  so  on  ;  how  far  will  it  fall  in  one  minute  ? 


444  pbogeessio:ss. 

833.  A  Geometrical  Progression  is  a  succes- 
sion of  numbers,  each  of  which  is  greater  or  less  than  the 
preceding  one  in  a  constant  ratio. 

Thus,  1,  3,  9,27,  81,  etc.,  is  a  geometrical  progression. 

834.  The  Terms  of  a  geometrical  progression  are 
the  numbers  of  which  the  progression  consists.  The  first 
and  last  terms  are  called  the  Extremes,  and  the  other 
terms  the  Means. 

835.  The  Ratio  of  a  geometrical  progression  is  the 
quotient  obtained  by  dividing  any  term  by  the  preceding 
one. 

836.  An  Increasing  Geometrical  Progres- 
sion is  one  in  which  the  ratio  is  greater  than  1. 

Thus,  1,  2,  4,  8,  16,  etc.,  is  an  increasing  progression. 

837.  A  Decreasing  Geometrical  Progres- 
sion is  one  in  which  the  ratio  is  less  than  1. 

Thus,  1,  |,  \,  $,  T^,  etc.,  is  a  decreasing  progression. 

838.  An  Infinite  Decreasing  Geometrical 
Progression  is  one  in  which  the  ratio  is  less  than  1, 
and  the  number  of  terms  infinite. 

Thus,  1,  \,  \,  \,  T^,  fa  ^4,  and  so  on  is  an  infinite  decreasing 
progression. 

839.  The  following  are  the  quantities  considered  in 
geometrical  progression  : 

1.  The  first  term  (a). 

2.  The  last  term   (I). 


3.  The  ratio  (r). 

4.  The  number  of  terms   (n). 


5.  The  sum  of  all  the  terms  (s). 


PROGRESSIONS.  445 

WRITTEN      EXERCISES, 

840.  To  find  one  of  the  extremes,  when  the  other 
extreme,  the  ratio,  and  the  number  of  terms  are 
given. 

1.  The  first  term  of  a  progression  is  2,  the  ratio  3,  and 
the  number  of  terms  10  ;  what  is  the  last  term  ? 

OPFUATTON 

Analysis.— The  2d  term  is  2  x  3  ;  the  third 
39=  19683  term  is  2x3x3  or  2  x  3* ;  the  4th  term  is 

2  2  x  33 ;   and  so  on.     Hence  Mie  10th  or  last 

oqqpfl 7         term  is  2  x  39  or  39366. 

2.  The  last  term  of  a  progression  is  393GG,  the  ratio  3, 
and  the  number  of  terms  10  ;  what  is"  the  first  term  ? 

operation.  Analysis.  —  The  first  term  must  be  a  num- 

39366  Der>  by  which  if  39  be  multiplied  the  product 

39      —  ^  =  a        Shan  be  39366  ;  hence,  if  39366  be  divided  by 
39,  the  quotient  will  be  the  first  term. 

3.  The  first  term  of  a  progression  is  1,  the  ratio  \,  and 
the  number  of  terms  9  ;  what  is  the  last  term  ? 

Eule. — I.  If  the  given  extreme  is  the  first  term,  multi- 
ply it  by  that  poiver  of  the  ratio  whose  exponent  is  one  less 
than  the  number  of  terms. 

II.  If  the  given  extreme  is  the  last  term,  divide  it  by 
that  power  of  the  ratio  whose  exponent  is  one  less  than  the 
number  of  terms. 

Formula.  — I  =  arn~x ;       a  =  — = . 

4.  The  first  term  of  a  geometrical  progression  is  6,  the 
ratio  4,  the  number  of  terms  6  ;  what  is  the  last  term  ? 

5.  The  last  term  is  192,  the  ratio  2,  and  the  number  of 
terms  7  ;  what  is  the  first  term  ? 


446  PROGRESSIONS. 

6.  A  drover  bought  20  cows,  agreeing  to  pay  $1  for  the 
first,  $2  for  the  second,  $4  for  the  third,  and  so  on  ;  how 
much  did  he  pay  for  the  last  cow  ? 

7.  Find  the  amount  of  $250  for  4  years  at  6  per  cent, 
compound  interest. 

The  first  term  is  250,  the  ratio  1.06,  and  the  number  of  terms  5. 

8.  If  1  cent  had  been  put  at  interest  in  1634,  what 
would  it  have  amounted  to  in  the  year  1874,  if  it  had 
doubled  its  value  every  12  years  ? 

841.  To  find  the  ratio,  when  the  extremes  and 
the  number  of  terms  are  given. 

1.  The  first  term  is  2,  the  last  term  512,  and  the  num- 
ber of  terms  5  ;  what  is  the  ratio  ? 

operation.  Analysts. — If  the  4th  power  of  the 

.si  3  —  256  ratio  be  multiplied  by  2,  the  product  will 

f/OKft  —  A—  be  513  (84°);  henCe'  if  513   be   divided 

V  —  4  _  r        by  ^  the  quotientf  256,  will  be  the  4th 

power  of  the  ratio.    Hence  the  ratio  is  the  4th  root  of  256,  or  4. 

2.  The  first  term  is  1,  the  last  term  -^,  and  the  num- 
ber of  terms  9  ;  what  is  the  ratio  ? 

Kule. — Divide  the  last  term  by  the  first,  and  extract 
that  root  of  the  quotient  whose  index  is  one  less  than  the 
number  of  terms. 


j~yi 


Formula. 

■    a 

3.  The  first  term  is  8,  the  last  term  5000,  and  the  num- 
ber of  terms  5  ;  what  is  the  ratio  ? 

4.  The  first  term  is  .0112,  the  last  term  7,  and  the 
number  of  terms  5  ;  what  is  the  ratio  ? 

5.  The  first  term  is  ^  the  last  term  15^,  and  the 
number  of  terms  7  ;  what  is  the  ratio  ? 


PROGRESSIONS.  447 

842.  To  find  the  number  of  terms,  when  the 
extremes  and  the  ratio  are  given. 

1.  The  extremes  are  2  and  512,  and  the  ratio  is  4  ;  what 

is  the  number  of  terms  ? 

operation.         Analysis. — If  512  be  divided  by  2,  the  quotient, 

2  )  512        256,  will  be  that  power  of  the  ratio  whose  exponent 

is  one  less  than  the  number  of  terms  (841).     But 

256  is  the  4th  power  of  the  ratio  4;  hence  the  num- 

4    =256       her  of  terms  is  5. 

2.  The  extremes  are  1  and  ^-J-g-,  and  the  ratio  is  | ;  what 
is  the  number  of  terms  ? 

Rule. — Divide  the  last  term  by  the  first ;  then  the  expo- 
nent of  the  power  to  ivhich  the  ratio  must  he  raised  to  jpro- 
duce  the  quotient  is  one  less  than  the  number  of  terms. 

Formula. — rn_1  =  - . 
a 

3.  The  extremes  are  2  and  1458,  and  the  ratio  is  3 ; 
what  is  the  number  of  terms  ? 

4.  The  extremes  are  -^  and  \,  and  the  ratio  2 ;  what 
is  the  number  of  terms  ? 

843.  To  find  the  sum  of  all  the  terms,  when  the 
extremes  and  the  ratio  are  given. 

1.  The  extremes  are  2  and  128,  and  the  ratio  is  4  ;  what 
is  the  sum  of  all  the  terms  ? 

OPEKATION. 

(128x4)-2_510 

4-1         -    3    --uu-s 

4  s  =       8  +  32  + 128  +  512  Analysis.— Subtract  the  sum  from  4 

8  —  2  +  8  +  32  +  128  times  the  sum,  and  510  remains,  which 

3  s  —  512  —  2  =  510  is  3  times  the  sum ;  hence,  ^-°-,or  170, 

lift  —  170  =  s  is  the  sum. 


448  PROGRESSIONS. 

2.  The  extremes  are  1  and  ^  and  the  ratio  is  \  ;  what 
is  the  sum  of  all  the  terms  ? 

6—i  +  i  +  i  +  i  +  A 

is=         i  +  i  +  j  +  iV  +  A 

Rule.—  Multiply  the  last  term  by  the  ratio,  and  divide 
the  difference  between  the  product  and  the  first  term  by  the 
difference  between  1  and  the  ratio. 

Formula. — s  = 7-. 

r  —  1 

3.  The  extremes  are  3  and  384,  and  the  ratio  is  2  ;  what 
is  the  sum  of  all  the  terms  ? 

4.  The  extremes  are  4|  and  T|T,  and  the  ratio  is  -J ; 
what  is  the  sum  of  all  the  terms  ? 

5.  What  is  the  sum  of  all  the  terms  of  the  infinite  pro- 
gression 8,  4,  2,  1,  J,  I,  ....  ? 

The  last  term  of  this  progression  may  be  conceived  as  0. 

6.  What  is  the  sum  of  all  the  terms  of  the  infinite  pro- 
gression 1,  £,  i,  ^V,  -gV,  .  .  .  .  ? 

7.  What  is  the  sum  of  1  -f  -J-+ J+-J-,  etc.,  to  infinity  ? 

8.  The  first  is  7,  the  ratio  3,  and  the  number  of  terms 
4  ;  what  is  the  sum  of  all  the  terms  ? 

First  find  the  last  term  by  Art.  840. 

9.  A  drover  bought  10  cows,  agreeing  to  pay  $1  for  the 
first,  $2  for  the  second,  M  for  the  third,  and  so  on  ;  what 
did  he  pay  for  the  10  cows  ? 

10.  If  a  man  were  to  buy  1 2  horses,  paying  2  cents  for 
the  first  horse,  6  cents  for  the  second,  and  so  on,  what 
would  they  cost  him  ? 


844.  An  Annuity  is  a  sum  of  money  payable  an- 
nually. The  term  is  also  applied  to  a  sum  of  money 
payable  at  any  equal  intervals  of  time. 

845.  A  Certain  Annuity  is  one  which  continues 
for  a  definite  period  of  time. 

846.  A  Perpetual  Annuity  or  Perpetuity 
is  one  which  continues  forever. 

847.  A  Contingent  Annuity  is  one  which  begins 
or  ends,  or  both  begins  and  ends,  on  the  occurrence  of 
some  specified  future  event  or  events. 

848.  An  Annuity  Forborne  or  in  Arrears 
is  one  the  payments  of  which  were  not  made  when  due. 

849.  The  Amount  or  Final  Value  of  an  an- 
nuity is  the  sum  of  all  the  payments  increased  by  the 
interest  of  each  payment  from  the  time  it  becomes  due 
until  the  annuity  ceases. 

850.  The  Present  Worth  of  an  annuity  is  such  a 
sum  of  money  as  will,  in  the  given  time,  and  at  the  given 
rate  per  cent.,  amount  to  the  final  value. 

851.  An  annuity  is  said  to  be  deferred  when  it  does 
not  begin  until  after  a  certain  period  of  time  ;  it  is  said 
to  be  reversionary  when  it  does  not  begin  until  after  the 
occurrence  of  some  specified  future  event,  as  the  death 
of  a  certain  person ;  and  it  is  said  to  be  in  possession 
when  it  has  begun,  or  begins  immediately. 


450  ANNUITIES. 

ANNUITIES    AT    SIMPLE    INTEREST. 

852,  All  problems  in  annuities  at  simple  interest  may 
be  solved  by  combining  the  rules  in  Arithmetical  Pro- 
gression with  those  in  Simple  Interest. 

WRITTEN    EXERCISES. 

853.  1 .  What  is  the  amount  of  an  annuity  of  $300  for 
5  years,  at  6  per  cent,  simple  interest  ? 

OPEEATION. 

300 -f- 372  Analysis— At  the  end  of  the  5th 
^ X  5  =  1680       year  the  following  sums  were  due  : 

The  5th  year's  payment  =  $300, 

The  4th  year's  payment  =  $300  +  $18  =  $318, 
The  3d  year's  payment  =  $300  +  $36  ==  $336, 
The  2d  year's  payment  =  $300  +  $54  ss  $354, 
The  1st  year's  payment  =  $300  +  $72  =  $372. 

These  sums  form  an  arithmetical  progression,  in  which  the  first 
term  is  the  annuity,  $300,  the  common  difference  is  the  interest  of 
the  annuity  for  1  year,  and  the  number  of  terms  is  the  number  of 
years.  The  sum  of  all  the  terms  of  this  progression  is  $1680  (8*32), 
which  is  the  amount  of  the  annuity. 

2.  A  father  deposits  annually  for  the  benefit  of  his  son, 
beginning  with  his  tenth  birthday,  such  a  sum  that  on 
his  21st  birthday  the  first  deposit,  at  simple  int.,  amounts 
to  $210,  and  the  sum  due  his  son  is  $1860.  Find  the 
annual  deposit,  and  at  what  rate  per  cent,  it  is  deposited. 

OPERATION. 

6  x  (1st  term  +  210)  ss  1860.     (832.) 

Hence,  1st  term  ss  310  —  210  =  100  ss  a. 

(210  —  100)  -fc  (12  -  1)  =  iff  =  10  =  d.     (830.) 

Analysis.— Here  $210,  the  first  deposit,  is  the  last  term  ;  12,  the 
number  of  deposits,  is  the  number  of  terms  ; 


ANNUITIES.  451 

and  $1860,  the  final  valne  of  the  annuity,  is  the  sum  of  all  the 
terms.  Using  the  principle  of  832,  we  find  the  first  term  to  be 
$100,  which  is  the  annual  deposit.  By  830,  the  common  dif- 
ference is  found  to  be  $10  ;  hence  10  per  cent,  is  the  required  rate. 

3.  What  is  the  amount  of  an  annuity  of  $150  for  5| 
years,  payable  quarterly,  at  1£  per  cent,  per  quarter  ? 

4.  What  is  the  present  worth  of  an  annuity  of  $300 
for  5  years,  at  6  per  cent.  ? 

5.  What  is  the  present  worth  of  an  annuity  of  $500 
for  10  years,  at  10  per  cent.  ? 

6.  In  what  time  will  an  annual  pension  of  $500  amount 
to  $3450,  at  6  per  cent,  simple  interest  ? 

7.  Find  the  rate  per  cent,  at  which  an  annuity  of  $6000 
will  amount  to  $59760  in  8  years,  at  simple  interest. 

8.  A  man  works  for  a  farmer  1  yr.  6  mo.,  at  $20  per 
month,  payable  monthly  ;  and  these  wages  remain  unpaid 
until  the  expiration  of  the  whole  term  of  service.  What 
is  due  the  workman,  allowing  simple  interest  at  6  per 
cent,  per  annum  ? 

ANNUITIES    AT    COMPOUND    INTEREST. 

854.  All  problems  in  annuities  at  compound  interest 
may  be  solved  by  combining  the  rules  in  Geometrical 
Progression  with  those  in  Compound  Interest. 

WRITTEN     EXERCISES. 

1.  What  is  the  amount  of  an  annuity  of  $300  for  5 
years,  at  6  per  cent,  compound  interest  ? 

OPERATION.  ANALYSTS.-At  the  end  of  the 

300  X  1.065— 300  _  5th  year  the    following  sums 

.06  are  due  : 


452  ANNUITIES. 

The  5th  year's  payment  l_ 

The  4th  year's  payment  +  interest  for  1  year  ss  $300  x  1.06, 

The  3d  year's  payment  +  compound  int.  for  2  years  =  $300  x  1.062, 
The  2d  year's  payment  +  compound  int.  for  3  years  =  $300  x  1.063, 
The  1st  year's  payment  +  compound  int.  for  4  years  =  $300  x  1.064. 

These  sums  form  a  geometrical  progression,  in  which  the  first 
term  is  the  annuity,  $300,  the  ratio  is  the  amount  of  $1  for  1  year, 
and  the  number  of  terms  is  the  number  of  years.  The  sum  of  all 
the  terms  of  this  progression  is  $1091.13  (843),  which  is  the 
amount  of  the  annuity. 

2.  What  is  the  present  worth  of  an  annuity  of  $300  for 
5  years,  at  6  per  cent,  compound  interest  ? 

operation..  Analysis.— The  amount  of  this  an- 

1691.13  nuity  is  $1691.13.    The  amount  of  $1  for 

T  mtf2'2P  ==  *  ^  years,  at  6  per  cent,  compound  interest, 

is  $1.338226(587).    Hence,  the  present 

worth  of  the  annuity  is  !faocxKm»  or  $1263.71. 

3.  Find  the  annuity  whose  amount  for  25  years,  at  6 
per  cent,  compound  interest,  is  $16459.35. 

4.  What  is  the  present  worth  of  an  annuity  of  $700 
for  7  years,  at  6  per  cent,  compound  interest  ? 

5.  An  annuity  of  $200  for  12  years  is  in  reversion  6 
years.  What  is  its  present  worth,  compound  interest 
at  6%? 

6.  A  man  bought  a  tract  of  land  for  $4800,  which  was 
to  be  pa;d  in  installments  of  $600  a  year ;  how  much 
uionoy;  at  6  per  cent,  compound  interest,  would  discharge 
Ihe  debt  at  the  time  of  the  purchase  ? 

7.  What  is  the  present  value  of  a  reversionary  lease  of 
£100,  commencing  14  years  hence,  and  to  continue  20 
fears,  compound  interest  at  5  per  cent.  ? 


REVIEW.  453 

S55.  SYNOPSIS  FOR  REVIEW. 

j_;    fj    Deps    \  *'  A  Power.  2.  Involution.  3.  Base,  or  Root.  4.  Ex- 
O  (       ponent.  5.  Square.  6.  Cube.  7.  Perfect  Power. 

H         2.  PRINCIPLE. 

S    ]   3.  802.    Rule.     1.  For  Integers.    2.  For  Fractions. 
^       4.  803.     1.  Principle.    2.  Geometrical  Illustration. 
£    I    5.  804.     1.  Principle.    2.  Geometrical  Illustration. 

1  Defs  i  ^  Square  R°ot-    2.  Cube  Root,  etc.    3.  Evolution. 
(      4.  Radical  Sign.     5.  Index. 

2.  810.  Rule. 

O       3.  812.  Principles,  1,  2,  3,  4. 

S       4.  813.  Rule,  I,  II,  III.    For  Fractions. 

,j  "S   5.  814.  Geometrical  Illustration. 

£       6.  818.  -Principles,  1,  2,  3,  4. 

W       7.  819.  Rule,  I,  II,  III,  IV,  V,  VI.    For  Fractions. 

8.  820.  Geometrical  Illustration. 

9.  822.  Roots  of  a  Higher  Degree.    Ride. 

( 1.  Arithmetical  Progression.    2.  Terms.    3.  Common 

1.  Defs.  <     Difference.  4.  Increasing  Arithmetical  Progression. 
(     5.  Decreasing  Arithmetical  Progression. 

2.  Quantities  considered. 

3.  829.    Rule,  I,  II.    Formulae 
m       4.  830.     Rule.    Formula, 

©  5.  831.     Rule.    Formula. 

m  6.  832.    Rule.    Formula. 

W  s                  / 1.  Geometrical  Progression.      2.  Terms.      3.  ifctfw. 

£  1.  Defs.  -J     4.  Increasing  Geom.  Prog.    5.  Decreasing  Geom. 

2  '      P?w.    6.  Infinite  Decreasina  Geom.  Proa. 


M 


iV<?#.    6.  Infinite  Decreasing  Geom.  Prog. 
pu       2.  Quantities  considered. 

3.  840.    Rule,  I,  II.    Formula. 

4.  841.     Rule.    Formula. 

5.  842.    Rule.    Formula. 

6.  843.    Rule.    Formula. 

f  1.  Annuity.    2.  Certain  Annuity.     3.  Perpetuity. 

4.  Contingent  Annuity.     5.  Annuity  in  Arrears. 

S    f  1.  Deps.  <       6.  Amount.     7.  Present  Worth  of  an  Annuity. 

£  8.  Deferred  Annuity.     9.  Reversionary  Annuity. 

^  <  [      10:  Annuity  in  Possession. 

5       3.  Annuities  at  Simple  Interest.  )  „    , .         .  ,     , 

K       *    .  ^  T  r  Problems,  how  solved. 

3    [  2.  Annuities  at  Comp.  Interest.  \  ' 


856.  Mensuration  is  the  process  of  finding  the  number  of 
units  in  extension. 

LINES. 

857.  A  Straight  Line  is  a  line  that 
• does  not  change  its  direction.     It  is  the  short- 
est distance  between  two  points. 

858.  A  Curved  Line  changes  its  direc- 
tion at  every  point. 

859.  Parallel  Lines  have  the  same 
direction  ;  and  being  in  the  same  plane  and 
equally  distant  from  each  other,  they  can  never 
meet. 

860.  A  Horizontal  Line  is  a  line  par- 
allel either  to  the  horizon  or  water  level. 

861.  A  Perpendicular  Line  is  a 
straight  line  drawn  to  meet  another  straight 
line,  so  as  to  incline  no  more  to  the  one  side 
than  to  the  other. 

A  perpendicular  to  a  horizontal  line  is  called  a  verti- 
cal line. 

ANGLES. 

862.  An  Angle  is  the  difference  in  the 
direction  of  two  lines  proceeding  from  a  com- 
mon point,  called  the  vertex. 

Angles  are  measured  by  degrees.    (301 .) 

863.  A  liight  Angle  is  an  angle  formed 
by  two  lines  perpendicular  to  each  other. 

N864.  An  Obtuse  Angle  is  greater  than 
a  right  angle. 
865.  An  Acute  Angle  is  less  than  a 
J right  angle. 

All  angles  except  right  angles  are  called  oblique  angles. 


Horizontal. 


TRIANGLES 


455 


PLANE  FIGURES. 

866.  A  Plane  Figure  is  a  portion  of  a  plane  surface  bounded 
by  straight  or  curved  lines. 

867.  A  Polygon  is  a  plane  figure  bounded  by  straight  lines. 

868.  The  Perimeter  of  a  polygon  is  the  sum  of  its  sides. 

869.  The  Area  of  a  plane  figure  is  the  surface  included 
within  the  lines  which  bound  it.    (460.) 

A  regular  polygon  has  all  its  sides  and  all  its  angles  equal. 

The  altitude  of  a  polygon  is  the  perpendicular  distance  between  its  base  and  a 
side  or  angle  opposite. 

A  polygon  of  three  sides  is  called  a  trigon,  or  triangle ;  of  four  sides,  a  tetra- 
gon,  or  quadrilateral ;  of  five  sides,  a  pentagon,  etc. 


Pentagon.       Hexagon.         Heptagon.        Octagon. 


Nonagon.         Decagon. 


TRIANGLES. 

870.  A  Triangle  is  a  plane  figure  bounded  by  three  sides, 
and  having  three  angles. 

871.  A  Right- Angled  Triangle 

is  a  triangle  having  one  right  angle. 

872.  The  Hypothenuse  of  aright- 
angled  triangle  is  the  side  opposite  the 
right  angle. 

873.  The  Base  of  a  triangle,  or  of 

any  plane  figure,  is  the  side  on  which  it  is  supposed  to  stand. 

874.  The  Perpendicular  of  a  right-angled  triangle  is  the 
side  which  forms  a  right  angle  with  the  base. 

875.  The  Altitude  of  a  triangle  is  a  line  drawn  from  the  angle 
opposite  perpendicular  to  the  base. 

1.  The  dotted  lines  in  the  following  figures  represent  the  altitude. 

2.  Triangles  are  named  from  the  relation  both  of  their  sides  and  angles. 


4:0(i  MEtfSUKATIOK. 

876.  An  Equilateral  Triangle  has  its  three  sides  equal. 

877.  An  Isosceles  Triangle  has  only  two  of  its  sides  equal. 

878.  A  Scalene  Triangle  has  all  of  its  sides  unequal. 
Fig.  1.  Fig.  2.  Fig.  3. 


Equilateral.  Isosceles.  Scalene. 

879.  An  Equiangular  Triangle  has  three  equal  angles 
(Kg.  1.) 

880.  An  Acute-angled  Triangle  has  three  acute  angles. 
(Fig.  2.) 

881.  An  Obtuse-angled  Triangle  has  one  obtuse  angle. 
(Fig.  3.) 

PROBLEMS. 

882.  The  base  and  altitude  of  a  triangle  being 
given  to  find  its  area. 

1.  Find  the  area  of  a  triangle  whose  base  is  26  ft.  and  altitude 
14.5  feet. 


14.5 


Operation.— 14.5  x  26-*-2=188£  sq.  ft.   Or,  26  x  —=188$  square 
feet,  area. 

2.  What  is  the  area  of  a  triangle  whose  altitude  is  10  yards  and 
base  40  feet? 
Rule. — 1.  Divide  the  product  of  the  base  and  altitude  by  2.    Or, 

2.  Multiply  the  base  by  one-half  the  altitude. 

Find  the  area  of  a  triangle 

3.  Whose  base  is  12  ft.  6  in.  and  altitude  6  ft.  9  in. 

4.  Whose  base  is  25.01  chains  and  altitude  18.14  chains. 

5.  What  is  the  cost  of  a  triangular  piece  of  land  whose  base  is 
15.48  ch.  and  altitude  9.67  ch.,  at  $60  an  acre? 

6.  At  $.40  a  square  yard,  find  the  cost  of  paving  a  triangular 
court,  its  base  being  105  feet,  and  its  altitude  21  yards  ? 

7.  Find  the  area  of  the  gable  end  of  a  house  that  is  28  ft.  wide, 
and  the  ridge  of  the  roof  15  ft.  higher  than  the  foot  of  the  rafters. 


TKIAHGLES.  457 

883.  The  area  and  one  dimension  being  given  to 
find  the  other  dimension. 

1.  What  is  the  base  of  a  triangle  whose  area  is  189  square  feet 
and  altitude  14  feet  ? 

Operation.— (189  sq.  ft.  x  2)-=- 14  =  27  ft.,  hose. 

2.  Find  the  altitude  of  a  triangle  whose  area  is  20*  square  feet 
and  base  3  yards. 

Rule. — Double  the  area,  then  divide  by  the  gioen  dimension. 

Find  the  other  dimension  of  the  triangle 

3.  When  the  area  is  65  sq.  in.  and  the  altitude  10  inches. 

4.  When  the  base  is  43  rods  and  the  area  588  sq.  rods. 

5.  When  the  area  is  6  £  acres  and  the  altitude  17  yards. 

6.  When  the  base  is  12.25  chains  and  the  area  5  A.  33  P. 

7.  Paid  $1050  for  a  piece  of  land  in  the  form  of  a  triangle,  at  the 
rate  of  %o\  per  square  rod.     If  the  base  is  8  rd.,  what  is  its  altitude  ? 

884.  The  three  sides  of  a  triangle  being  given  to 
find  its  area. 

1.  Find  the  area  of  a  triangle  whose  sides  are  30,  40,  and  50  ft. 

Operation.— (30  +  40  +  50)-*-2  =  60;  60-30  =  30;  60-40  =  20; 
60-50  =  10.     y'eOx 30 x 20 x  10  =  600  ft.,  area. 

2.  What  is  the  area  of  an  isosceles  triangle  whose  base  is  20  ft., 
and  each  of  its  equal  sides  15  feet  ? 

Rule.— From  half  the  sum  of  the  three  sides,  subtract  each  side 
separately  ;  multiply  the  half -sum  and  the  three  remainders  together ; 
the  square  root  of  the  product  is  the  area. 

3.  Find  the  area  of  a  triangle  whose  sides  are  25,  36,  and  49  in. 

4.  How  many  acres  in  a  field  in  the  form  of  an  equilateral  tri- 
angle whose  sides  each  measure  70  rods  ? 

5.  The  roof  of  a  house  30  ft.  wide  has  the  rafters  on  one  side 
20  ft.  long,  and  on  the  other  18  ft.  long.  How  many  square  feet  of 
boards  will  be  required  to  board  up  both  gable  ends  ? 


458 


MENSURATION. 


885.  The  following  principles  relating  to  right-angled  triangles 
have  been  established  by  Geometry  : 

Principles.— 1.  The  square  of  the 
hypothenuse  of  a  right-angled  triangle 
is  equal  to  the  sum  of  the  squares  of 
the  other  tico  sides. 

2.  The  square  of  the  base,  or  of  the 
perpendicular,  of  a  right-angled  tri- 
angle is  equal  to  the  square  of  the 
hypothenuse  diminished  by  the  square 
of  the  other  side. 


886.  To  find  the  hypothenuse.    • 

1.  The  base  of  a  right-angled  triangle  is  12,  and  the  perpendicu- 
lar 16.    What  is  the  length  of  the  hypothenuse  ? 

Operation.— 122  + 162  =  400  (Prin.  1).    ^m  =  20,  hypothenuse. 

2.  The  foot  of  a  ladder  is  15  feet  from  the  base  of  a  building,  and 
the  top  reaches  a  window  36  feet  above  the  base.  What  is  the 
length  of  the  ladder  ? 

Rule. — Extract  the  square  root  of  the  sum  of  the  squares  of  the 
base  and  the  perpendicular  ;  the  result  is  the  hypothenuse. 

3.  If  the  gable  end  of  a  house  40  ft.  wide  is  16  ft.  high,  what  is 
the  length  of  the  rafters  ? 

4.  A  park  25  chains  long  and  23  chains  wide  has  a  walk  running 
through  it  from  opposite  corners  in  a  straight  line.  What  is  the 
length  of  the  walk  ? 

5.  A  room  is  20  ft.  long,  16  ft.  wide,  and  12  ft.  high  What  is  the 
distance  from  one  of  the  lower  corners  to  the  opposite  upper  corner  ? 

887.  To  find  the  base  or  perpendicular. 

1.  The  hypothenuse  of  a  right-angled  triangle  is  35  feet,  and  the 
perpendicular  28  feet.     Find  the  base. 


Operation.— 35s  -  28'  =  441  (Prin.  2).    ^441  =  21  ft.,  base. 


QUADRILATEEALS. 


459 


2.  The  hypothenuse  of  a  right-angled  triangle  is  53  yards  and 
the  base  84  feet.     Find  the  perpendicular. 

•Rule.—  Extract  the  square  root  of  the  difference  between  the  square 
of  the  hypothenuse  and  the  square  of  the  given  side  ;  the  result  is  the 
required  side. 

3.  Find  the  width  of  a  house,  whose  rafters  are  12  ft.  and  15  ft. 
long,  and  that  form  a  right  angle  at  the  point  in  which  they  meet. 

4.  A  line  reaching  from  the  top  of  a  precipice  120  feet  high,  on 
the  bank  of  a  river,  to  the  opposite  side  is  380  feet  long.  How 
wide  is  the  river  ? 

5.  A  ]  adder  52  ft.  long  stands  against  the  side  of  a  building. 
I!  ow  many  feet  must  it  be  drawn  out  at  the  bottom  that  the  top 
may  be  lowered  4  feet  ? 

QUADRILATERALS. 

888.  A  Quadrilateral  is  a  plane  figure  bounded  by  four 
straight  lines. 

There  are  three  kinds  of  quadrilaterals,  the  Parallelogram,  Trapezoid,  and 
Trapezium. 

889.  A  Parallelogram  is  a  quadrilateral  which  has  its 
opposite  sides  parallel. 

There  are  four  kinds  of  parallelograms,  the  Square,  Rectangle,  Rhomboid,  and 
Rhombus. 

890.  A  Rectangle  is  any  parallelogram  having  its  angles 
right  angles. 

891.  A  Square  is  a  rectangle  whose  sides  are  equal. 

892.  A  Rhomboid  is  a  parallelogram  whose  opposite  sides 
only  are  equal,  and  whose  angles  are  not  right  angles. 

893.  A  Rhombus  is  a  parallelogram  whose  sides  are  all 
equal,  but  whose  angles  are  not  right  angles. 


Square. 


Rectangle. 


Rhomboid. 


Rhombus. 


460 


MENSURATION 


894.  A  Trapezoid  is  a  quadrilateral,  two  of  whose  sides  are 
parallel. 

895.  A  Trapezium  is  a  quadrilateral  having  no  two  sides 
parallel. 

896.  The  Altitude  of  a  parallelogram  or  of  a  trapezoid  is  the 
perpendicular  distance  between  its  parallel  sides. 

The  clotted  vertical  lines  in  the  figure  represent  the  altitude. 

897.  A  Diagonal  of  a  plane  figure  is  a  straight  line  joining 
the  vertices  of  two  angles  not  adjacent. 


Parallelogram. 


Trapezoid. 


Trapezium. 


PJi  O  BLEMS  . 

898.  To  find  the  area  of  any  parallelogram. 

1.  Find  the  area  of  a  parallelogram  whose  base  is  16.25  feet  and 
altitude  7.5  feet. 

Operation.— 16.25  ft.  x  7.5  =  121.875  sq.  feet,  area. 

2.  The  base  of  a  rhombus  is  10  feet  6  inches,  and  its  altitude 
8  feet.     What  is  its  area  ? 

Rule. — Multiply  the  base  by  ths  altitude. 

3.  How  many  acres  in  a  piece  of  land  in  the  form  of  a  rhomboid, 
the  base  being  8.75  ch.  and  altitude  6  chains  ? 

899.  To  find  the  area  of  a  trapezoid. 

1.  Find  the  area  of  a  trapezoid  whose  parallel  sides  are  23  and 
11  feet,  and  the  altitude  9  feet. 


Operation.— 23  ft.  + 11  ft.-^2=17  ft. ;  17  ft.  x  9=153  sq.  ft.,  area. 

2.  Required  the  area  of  a  trapezoid  whose  parallel  sides  are  178 
and  146  feet,  and  the  altitude  69  feet. 

Rule.— Multiply  one-  half  the  sum  of  the  parallel  sides  by  the 
altitude. 


C  I  E  C  L  E  S  .  4G1 

3.  How  many  square  feet  in  a  board  16  ft.  long,  18  incbfcs  wide 
at  one  end  and  25  inches  wide  at  the  other  end  ? 

4.  One  side  of  a  quadrilateral  field  measures  88  rods  ;  the  side 
opposite  and  parallel  to  it  measures  26  rods,  and  the  distance  be- 
tween the  two  sides  is  10  rods.     Find  the  area. 

900.  To  find  the  area  of  a  trapezium. 

1.  Find  the  area  of  a  trapezium  whose  ^y^^vks. 

diagonal  is  42  feet  and  perpendiculars  to  this  ^-^""^   4'2  ft-  liS     j\ 

diagonal,  as  in  the  diagram,  are  16  feet  and  \.      d]  s' 

18  feet.  \^  y^ 


Opeeation.— (18  ft.  + 16  ft.  -*-2)  x  42  =  714  sq.  feet,  area. 

2.  Find  the  area  of  a  trapezium  whose  diagonal  is  35  ft.  6  in.,  and 
the  perpendiculars  to  this  diagonal  9  feet  and  3  feet. 

Rule. — Multiply  the  diagonal  by  half  the  sum  of  the  perpendicu- 
lars drawn  to  it  from  the  vertices  of  opposite  angles. 

3.  How  many  acres  in  a  quadrilateral  field  whose  diagonal  is 
80  rd.  and  the  perpendiculars  to  this  diagonal  20.453  and  50.832  rd.  ? 

To  find  the  area  of  any  regular  polygon,  multiply  its  perimeter,  or  the  sum  of 
its  sides,  by  one-half  the  perpendicular  falling  from  its  center  to  one  of  its  sides. 

To  find  the  area  of  an  irregular  polygon,  divide  the  figure  into  triangles  and 
trapeziums,  and  find  the  area  of  each  separately.  The  sum  of  these  areas  will 
be  the  area  of  the  whole  polygon. 


THE  CIRCLE. 

901.  A  Circle  is  a  plane  figure  bounded  by  a  curved  line, 
called  the  circumference,  every  point  of  which  is 
equally  distant  from  a  point  within  called  the 
center. 


902.  The  Diameter  of  a  circle  is  a  line 
passing  through  its  center,  and  terminated  at  both 
ends  by  the  circumference. 

903.  The  Radius  of  a  circle  is  a  line  extending  from  its  cen- 
ter to  any  point  in  the  circumference.     It  is  one-half  the  diameter. 


462  MENSURATION. 

PROBLEMS. 

904.  When  either  the  diameter  or  the  circum- 
ference of  a  circle  is  given,  to  find  the  other  di- 
lneiision^of  it. 

1.  Find  the  circumference  of  a  circle  whose  diameter  is  20  inches. 
Operation.— 20  in.  x  3.1416  =  62.802  in.  =  5  ft.  2.802  in.,  circum. 

2.  Find  the  diameter  of  a  circle  whose  circumference  is  62.832  ft 
Operation.— 62.832  ft.-f-3.1416  =  20  ft.,  diameter. 

3.  Find  the  diameter  of  a  wheel  whose  circumference  is  50  feet. 

Rule. — 1.  Multiply  the  diameter  by  3.1416  ;  the  product  is  the  cir- 
cumference. 
2.  Divide  the  circumference  by  3.1416  ;  the  quotient  is  the  diameter. 

4.  What  is  the  diameter  of  a  tree  whose  girt  is  18  ft.  6  in.  ? 

5.  What  is  the  radius  of  a  circle  whose  circumference  is  31.416  ft.? 

6.  Find  the  circumference  of  the  greatest  circle  that  can  be 
drawn  with  a  string  14  inches  long,  used  as  a  radius. 

905.  To  find  the  area  of  a  circle,  when  both  its 
diameter  and  circumference  are  given,  or  when 
either  is  given. 

1.  What  is  the  area  of  a  circle  whose  diameter  is  10  feet  and  cir- 
cumference 31.416  feet? 


Operation.— 31.416  ft.  x  10-=-4  =  78.54  sq.  ft.,  area. 

2.  Find  the  area  of  a  circle  whose  diameter  is  10  feet. 
Operation.— 10  ft.2  x  .7854  =  78.54  sq.  feet,  area. 

3.  Find  the  area  of  a  circle  whose  circumference  is  31.416  feet. 
Operation.— 31.416  ft.-f-3.l416=10  ft.,  diam.;  (10  ft.)2  x  .7854= 

78.54  sq.  feet,  area. 

Rules.— To  find  the  area  of  a  circle  : 

1.  Multiply  \  of  its  diameter  by  the  circumference. 

2.  Multiply  the  square  of  its  diameter  by  .7854. 

4.  What  is  the  area  of  a  circular  pond  whose  circumference  is 
200  chains  ? 

5.  The  distance  around  a  circular  park  is  1)  miles.     How  many 
acres  does  it  contain  ? 


CIRCLES.  463 

906.  To  find  the  diameter  or  the  circumference 
of  a  circle,  when  the  area  is  given. 

1.  What  is  the  diameter  of  a  circle  whose  area  is  1319.472  ? 

Operation— 1319. 472-T-.7854  =  1680  ;  ^1680  =  40.987  +  ,  diam- 
eter. 

2.  What  is  the  circumference  of  a  circle  whose  area  is  19.635  ? 

Operation— 19.635  -f-  3.1416  =  6.25  ;  -y/6\25=2.5,  radius;  2.5  x 
2x3.1416  =  15.708,  circumference. 

Rule. — 1.  Divide  the  area  by  .7854  and  extract  the  square  root  of 
the  quotient ;  the  result  is  the  diameter. 

2.  Divide  the  area  by  3.1416  and  extract  the  square  root  of  the 
quotient ;  the  result  is  the  radius.  The  circumference  is  obtained  by 
Art.  904,  Or, 

3.  Divide  the  area  by  .07958  and  find  the  square  root  of  the  quotient. 

3.  The  area  of  a  circular  lot  is  38.4846  square  rods.  What  is  its 
diameter  ? 

4.  The  area  of  a  circle  is  286.488  square  feet.  Required  the 
diameter  and  the  circumference. 

907.  To  find  the  side  of  an  inscribed  square  when 
the  diameter  of  the  circle  is  known. 

1.  What  is  the  side  of  a  square  inscribed  in  a 
circle  whose  diameter  is  6  rods  ? 

Operation.— 62  -4-  2  =  18 ;  ^18=4.24  rods,  side 
of  square. 

2.  The  diameter  of  a  circle  is  200  feet.    Find 
the  side  of  the  inscribed  square. 

Rule. — 1.  Extract  the  square  root  of  half  the  square  of  the  diam- 
eter    Or, 

2.  Multiply  the  diameter  by  .7071. 

3.  The  circumference  of  a  circle  is  104  yards.  Find  the  side  of 
the  inscribed  square. 

4.  The  area  of  a  circle  is  78.54  square  feet.  Find  the  side  of  the 
inscribed  square. 


464  ME^SUKATIOK. 

908.  To  find  the  area  of  a  circular  ring   formed 
by  two  concentric  circles. 

1.  Find  the  area  of  a  circular  ring,  when 
the  diameters  of  the  circles  are  20  and  30  feet. 


Operation.— (30  -f  20  x  30  -  20)  x  .7854  = 
292.7  sq.  ft.,  area. 

2.  Find  the  area  of  a  circular  ring  formed 
by  two  concentric  circles,  whose  diameters  are 
7  ft.  9  in.  and  4  ft.  3  in. 
Rule. — Multiply  the  sum  of  the  two  diameters  by  their  difference, 
and  the  product  by  .7854 ;  the  result  is  the  area. 

3.  Two  diameters  are  35.75  and  16.25  ft. ;  find  the  area  of  the  ring. 

4.  The  area  of  a  circle  is  1  A.  154.16  P.    In  the  center  is  a  pond  of 
water  10  rd.  in  diameter  ;  find  the  area  of  the  land  and  of  the  water. 

909.  To  find  a  mean  proportional  between  two 
numbers. 

1.  What  is  a  mean  proportional  between  3  and  12  ? 


Operation. — ^12  x  3  =  6,  the  mean  proportional. 

When  three  numbers  are  proportional,  the  product  of  the  extremes  is  equal  to 
the  square  of  the  mean. 

Rule. — Extract  the  square  root  of  the  product  of  the  two  numbers. 

Find  a  mean  proportional  between 

2.  42  and  168.        |        3.  64  and  12.25.  4.  ff  and  ¥V 

5.  A  tub  of  butter  weighed  36  lb.  by  the  grocer's  scales  ;  but 
weighing  it  in  the  other  scale  of  the  balance,  it  weighed  only  30 
pounds.     What  was  the  true  weight  of  the  butter  ? 

SIMILAR   PLANE   FIGURES. 

910.  Similar  Plane  Figures  are  such  as  have  the  same 
form,  viz.,  equal  angles,  and  their  like  dimensions  proportional. 

All  circles,  squares,  equiangular  triangles,  and  regular  polygons  of  the  same 
number  of  sides  are  similar  figures. 
The  like  dimensions  of  circles  are  their  radii,  diameters,  and  circumferences. 

Principles. — 1.  The  like  dimensions  of  similar  plane  figures  are 
proportional. 


SIMILAR     PLANE     FIGURES.  465 

2.  The  areas  of  similar  plane  figures  are  to  each  other  as  the  squares 
of  their  like  dimensions.     And  conversely, 

3.  The  like  dimensions  of  similar  plane  figures  are  to  each  oilier  as 
the  square  roots  of  their  areas. 

The  same  principles  apply  also  to  the  surfaces  of  all  similar  figures,  such  as 
triangles,  rectangles,  etc. ;  the  surfaces  of  similar  solids,  as  cubes,  pyramids,  etc.; 
and  to  similar  curved  surfaces,  as  of  cylinders,  cones,  and  spheres.    Hence, 

4.  TJie  surfaces  of  all  similar  figures  are  to  each  other  as  the  squares 
oftlieir  like  dimensions.    And  conversely, 

5.  Their  dimensions  are  as  the  square  roots  of  their  surfaces. 

PROBLEMS. 

1.  A  triangular  field  whose  base  is  12  cli.  contains  2  A.  80  P. 
Find  the  area  of  a  field  of  similar  form  whose  base  is  48  chains. 

Operation.— 122 :  48* : :  2  A.  80  P. :  x  P. =6400  P.  =  40  A.,  area, 
(Prin.  2.) 

2.  The  side  of  a  square  field  containing  18  acres  is  60  rods  long. 
Find  the  side  of  a  similar  field  that  contains  ^  as  many  acres. 

Operation.— 18  A.  :  6  A.  : :  602  :  a2  =1200  ;  ^/vm  =  34.64  rd.  + , 
side.  '  (Prin.  3.) 

3.  Two  circles  are  to  each  other  as  9  to  16  ;  the  diameter  of  the 
less  being  112  feet,  what  is  the  diameter  of  the  greater? 

Operation.— 9  :  16  : :  1122 :  x*  =  3  :  4  : :  112  :  x  =  149  ft.  4  in., 
diameter.    (Prin.  2.) 

4.  A  peach  orchard  contains  720  square  rods,  and  its  length  is  to 
Its  breadth  as  5  to  4  ;  what  are  its  dimensions  ? 

Operation.— The  area  of  a  rectangle  5  by  4  equals  20  (898). 

20  :  720  :  :  52  :  z2  =  900  ;     ^900  =  30  rd.,  length. 

20  :  720  : :  42  :  z2  =  576  ;     ^576  =  24  rd.,  width. 

5.  It  is  required  to  lay  out  283  A.  107  P.  of  land  in  the  form  of 
a  rectangle,  so  that  the  length  shall  be  3  times  the  width.  Find 
the  dimensions. 

6.  A  pipe  1.5  in.  in  diameter  fills  a  cistern  in  5  hours ;  find  the 
diameter  of  a  pipe  that  will  fill  the  same  cistern  in  55  min.  6  sec. 

7.  The  area  of  a  triangle  is  24276  sq.  ft.,  and  its  sides  in  proportion 
to  the  numbers  13,  14,  and  15.     Find  the  length  of  its  sides  in  feet. 


466  MENSURATION. 

8.  If  it  cost  $167.70  to  enclose  a  circular  pond  containing  17  A. 
110  P.,  what  will  it  cost  to  enclose  another  ^  as  large  ? 

9.  If  63.39  rods  of  fence  will  enclose  a  circular  field  containing 
2  acres,  what  length  will  enclose  8  acres  in  circular  form  ? 

REVIEW    OF    PLANE    FIGURES. 

PROBLEMS. 

911.  1.  How  much  less  will  the  fencing  of  20  acres  cost  in  the 
square  form  than  in  the  form  of  a  rectangle  whose  breadth  is  \  the 
length,  the  price  being  $2.40  per  rod  ? 

2.  A  house  that  is  50  feet  long  and  40  feet  wide  has  a  square  or 
pyramidal  roof,  whose  height  is  15  ft.  Find  the  length  of  a  rafter 
reaching  from  a  corner  of  the  building  to  the  vertex  of  the  roof. 

3.  Find  the  diameter  of  a  circular  island  containing  1\  sq.  miles. 

4.  What  is  the  value  of  a  farm,  at  $75  an  acre,  its  form  being  a 
quadrilateral,  with  two  of  its  opposite  sides  parallel,  one  40  ch. 
and  the  other  22  ch.  long,  and  the  perpendicular  distance  between 
them  25  chains  ? 

5.  Find  the  cost,  at  18  cents  a  square  foot,  of  paving  a  space  in 
the  form  of  a  rhombus,  the  sides  of  which  are  15  feet,  and  a  per- 
pendicular drawn  from  one  oblique  angle  will  meet  the  opposite 
side  9  feet  from  the  adjacent  angle. 

6.  A  goat  is  fastened  to  the  top  of  a  post  4  ft.  high  by  a  rope  50  ft. 
long.     Find  the  area  of  the  greatest  circle  over  which  he  can  graze. 

7.  How  much  larger  is  a  square  circumscribing  a  circle  40  rods 
in  diameter,  than  a  square  inscribed  in  the  same  circle  ? 

8.  What  is  the  value  of  a  piece  of  land  in  the  form  of  a  triangle, 
whose  sides  are  40,  48,  and  54  rods,  respectively,  at  the  rate  of 
$125  an  acre  ? 

9  The  radius  of  a  circle  is  5  feet ;  find  the  diameter  of  another 
circle  containing  4  times  the  area  of  the  first. 

10.  How  many  acres  in  a  semi-circular  farm,  whose  radius  is 
100  rods  ? 

11.  What  must  be  the  width  of  a  walk  extending  around  a  gar- 
den 100  feet  square,  to  occupy  one-half  the  ground? 

12.  An  irregular  piece  of  land,  containing  540  A.  36  P.  is  ex- 
changed for  a  square  piece  of  the  same  area  ;  find  the  length  of  one 
of  its  sides  ?  If  divided  into  42  equal  squares,  what  is  the  length 
of  the  side  of  each  ? 


SOLIDS 


46? 


13.  A  field  containing  15  A.  is  30  rd.  wide,  and  is  a  plane  inclining 
in  the  direction  of  its  length,  one  end  being  120  ft.  higher  than  the 
other.     Find  how  many  acres  of  horizontal  surface  it  contains. 

14  If  a  pipe  3  inches  in  diameter  discharges  12  hogsheads  of 
water  in  a  certain  time,  what  must  be  the  diameter  of  a  pipe  which 
will  discharge  48  hogsheads  in  the  same  time  ? 

SOLIDS. 

912.  A  Solid  or  Body  has  three  dimensions,  length,  breadth, 
and  thickness. 

The  planes  which  bound  it  are  called  its  faces,  and  their  intersections,  its 
edges. 

913.  A  Prism  is  a  solid  whose  ends  are  equal  and  parallel, 
similar  polygons,  and  its  sides  parallelograms. 

Plifrma  take  their  names  from  the  form  of  their  bases,  as  triangular,  quad- 
rangular, pentagonal,  etc 

914.  The  Altitude  of  a  prism  is  the 
perpendicular  distance  between  its  bases. 

915.  A    Parallelopipedon    is    a 

prism  bounded  by  six  parallelograms,  the 
opposite  ones  being  parallel. 

916.  A  Cube  is  a  parallelopipedon 
whose  faces  are  all  equal  squares. 

917.  A  Cylinder  is  a  body  bounded  Cube. 

by  a  uniformly   curved  surface,  its  ends  being  equal  and  parallel 
circles. 

1.  A  cylinder  is  conceived  to  be  generated  by  the  revolution  of  a  rectangle 
about  one  of  its  sides  as  an  axis. 

2.  The  line  joining  the  centers  of  the  bases,  or  ends,  of  the  cylinder  is  its  alti- 
tude, or  axis. 


Pentagonal 
Prism- 


Cylinder. 


¥68 


MEKSUKATION. 


P  It  O  B  ZJEMS. 

918.  To  find  the  convex  surface  of  a  prism  or 
cylinder. 

1.  Find  the  area  of  the  convex  sur- 
face of  a  prism  whose  altitude  is  7  ft., 
and  its  base  a  pentagon,  each  side  of 
which  is  4  feet. 

Operation.— 4  ft.  x  5  =  20  ft.,  peri- 
meter. 
20  ft.  x  7=140  sq.  ft.,  convex  surface. 

2.  Find  the  area  of  the  convex  sur- 
face of  a  triangular  prism,  whose  alti- 
tude is  8 1  feet,  and  the  sides  of  its  base 
4,  5,  and  6  feet,  respectively. 

Operation.  —4  ft.  +  5  f t.  +  6  ft.  = 
15  ft.,  perimeter. 
15  ft.  x  8£=127|  sq.  ft.,  convex  surf  ace. 

3.  Find  the  area  of  the  convex  surface  of  a  cylinder  whose  altitude 
is  2  ft.  5  in.  and  the  circumference  of 
its  base  4  ft.  9  in. 

Operation.— 2  ft.  5  in. =29  in. ;  4  ft. 
9  in.  =  57  in. 

57  in.  x  29  =  1653  sq.  in.  =  11  sq.  ft 
69  sq.  inches,  convex  surface. 

Rule. — Multiply  the  perimeter  of  the  base  by  the  altitude. 
To  find  the  entire  surface,  add  the  area  of  the  hases  or  ends. 

4.  If  a  gate  8  ft.  high  and  6  ft.  wide  revolves  upon  a  point  in  its 
center,  what  is  the  entire  surface  of  the  cylinder  described  by  it  ? 

5.  Find  the  superficial  contents,  or  entire  surface  of  a  parallelo- 
pipedon  8  ft.  9  in.  long,  4  ft.  8  in.  wide,  and  3  ft.  3  in.  high. 

6.  What  is  the  entire  surface  of  a  cylinder  formed  by  the  revo- 
lution about  one  of  its  sides  of  a  rectangle  that  is  6  ft.  6  in.  long 
and  4  ft.  wide  ? 

7.  Find  the  entire  surface  of  a  prism  whose  base  is  an  equilateral 
triangle,  the  perimeter  being  18  ft.,  and  the  altitude  15  ft. 


PYRAMIDS     AND     CONES. 


469 


919.  To  find  the  volume  of  any  prism  or  cylinder. 

1.  Find  the  volume  of  a  triangular  prism,  whose  altitude  is  20  ft., 
and  each  side  of  the  base  4  feet. 

Opekation.— The  area  of  the  base  is  6.928  sq.  ft.  (882). 
6.928  sq.  ft.  x  20  =  138.56  cu.  ft.,  volume. 

2.  Find  the  volume  of  a  cylinder  whose  altitude  is  8  ft.  6  in.,  and 
the  diameter  of  its  base  3  feet. 

Operation.— 32  x  .7854  =  7.0686  square  feet,  area  of  base  (905) 
7.0686  sq.  ft.  x  8.5  =  60.083  cubic  feet,  volume. 
Rule. — Multiply  the  area  of  the  base  by  the  altitude. 

3.  Find  the  solid  contents  of  a  cube  whose  edges  are  6  ft.  6  in. 

4.  Find  the  cost  of  a  piece  of  timber  18  in.  square  and  40  ft.  long, 
at  $.30  a  cubic  foot. 

5.  Required  the  solid  contents  of  a  cylinder  whose  altitude  is 
15  ft.  and  its  radius  1  ft.  3  in. 

6.  What  is  the  value  of  a  log  24  ft.  long,  of  the  average  circum- 
ference of  7.9  ft.,  at  $.45  a  cubic  foot  ? 


PYRAMIDS    AND    CONES. 

920.  A  Pyramid  is  a  body  having  for  its  base  a  polygon, 
and  for  its  other  faces  three  or  more  triangles,  which  terminate  in 
a  common  point  called  the  vertex. 

Pyramids,  like  prisms,  take  their  names  from  their  bases,  and  are  called  tri- 
angular, square,  or  quadrangular,  pentagonal,  etc. 


Cone. 


Frustum. 


Pyramid.  Frustum. 

921.  A  Cone  is  a  body  having  a  circular  base,  and  whose  con- 
vex surface  tapers  uniformly  to  the  vertex. 

It  is  a  body  conceived  to  be  formed  by  the  revolution  of  a  right  angled  triangle 
about  one  of  its  sides  containing  the  right  angle,  as  an  immovable  axis. 

922.  The  Altitude  of  a  pyramid  or  of  a  cone  is  the  perpendic- 
ular distance  from  its  vertex  to  the  plane  of  its  base. 


470  MENSURATION. 

923.  The  Slant  Height  of  &  pyramid  is  the  perpendicular  dis- 
tance from  its  vertex  to  one  of  the  sides  of  the  base  ;  of  a  cone,  is  a 
straight  line  from  the  vertex  to  the  circumference  of  the  base. 

924.  The  Frustum  of  a  pyramid  or  of  a  cone  is  that  part 
which  remains  after  cutting  off  the  top  by  a  plane  parallel  to  the 
base. 

PROBLEMS. 

925.  To  find  the  convex  surface  of  a  pyramid  or 
of  a  cone. 

1.  Find  the  convex  surface  of  a  triangular  pyramid,  the  slant 
height  being  16  ft.,  and  each  side  of  the  base  5  feet. 

Operation.— (5  ft. +  5  ft.  +5  ft.)  x  16h-2  =  120  sq.  ft.,  conv.  surf. 

2.  Find  the  convex  surface  of  a  cone  whose  diameter  is  17  ft.  6  in., 
and  the  slant  height  30  feet. 

Rule. — Multiply  the  perimeter  or  circumference  of  the  base  by  one- 
half  the  slant  height. 
To  find  the  entire  surface,  add  to  this  product  the  area  of  the  hase. 

3.  Find  the  entire  surface  of  a  square  pyramid  whose  base  is  8  ft. 
6  in.  square,  and  its  slant  height  21  feet. 

4.  Find  the  entire  surface  of  a  cone  the  diameter  of  whose  base 
is  6  ft.  9  in.  and  the  slant  height  45  ft. 

5.  Find  the  cost  of  painting  a  church  spire,  at  $.25  a  sq.  yd.,  whose 
base  is  a  hexagon  5  ft.  on  each  side,  and  the  slant  height  GO  feet. 

926.  To  find  the  volume  of  a  pyramid  or  of  a  cone. 

1.  What  is  the  volume,  or  solid  contents,  of  a  square  pyramid 
whose  base  is  6  feet  on  each  side,  and  its  altitude  12  feet  ? 


Operation.— G  x  6  x  12  -f-  3  =  144  cu.  ft.,  volume. 

2.  Find  the  volume  of  a  cone,  the  diameter  of  whose  base  is  5  ft. 
and  its  altitude  10 1  feet? 

Operation.— (52  ft.  x  .7854)  x  10JT3  =  68.72^  cu.  ft.,  volume. 
Rule. — Multiply  the  area  of  the  base  by  one-third  the  altitude. 

3.  Find  the  solid  contents  of  a  cone  whose  altitude  is  24  ft. ,  and 
the  diameter  of  its  base  30  inches. 

4.  What  is  the  cost  of  a  triangular  pyramid  of  marble,  whose 
altitude  is  9  ft.,  each  side  of  the  base  being  3  ft.,  at  $2^  per  cu.  foot  ? 

5.  Find  the  volume  and  the  entire  surface  of  a  pyramid  whose 
base  is  a  rectangle  80  feet  by  60  feet,  and  the  edges  which  meet  at 
the  vertex  are  130  feet. 


PYRAMIDS     AND     CO^ES.  471 

927.  To  find  the  convex  surface  of  a  frustum  of  a 
pyramid  or  of  a  cone. 

1.  What  is  the  convex  surface  of  a  frustum  of  a  square  pyramid, 
whose  slant  height  is  7  feet,  each  side  of  the  greater  base  4  feet,  and 
of  the  less  base  18  inches? 

Operation. — The  perimeter  of  the  greater  base  is  16  ft.,  of  the  less 

6  feet. 

16  ft. +  6  ft.  x  7  -*-2=  77  sq.  ft.,  convex  surface. 

2.  Find  the  convex  surface  of  a  frustum  of  a  cone  whose  slant 
height  is  15  feet,  the  circumference  of  the  lower  base  30  feet,  and 
of  the  upper  base  16  feet. 

Rule. — Multiply  the  sum  of  the  perimeters,  or  of  ihg  circumfer- 
ences, by  one-half  the  slant  height. 
To  find  the  entire  surface,  add  to  this  product  the  area  of  both  ends,  or  bases. 

3.  How  many  square  yards  in  the  convex  surface  of  a  frustum 
of  a  pyramid,  whose  bases  are  heptagons,  each  side  of  the  lower 
base  being  8  feet,  and  of  the  upper  base  4  feet,  and  the  slant  height 
55  feet? 

928.  To  find  the  volume  of  a  frustum  of  a  pyramid 
pyramid  or  of  a  cone. 

1.  Find  the  volume  of  the  frustum  of  a  square  pyramid  whose 
altitude  is  10  feet,  each  side  of  the  lower  base  12  feet,  and  of  the 
upper  base  9  feet. 


Operation.— 12'  + 95  =  225  ;  (225  +  ^144x81)  x  10-^3=1110  cu. 
feet,  volume. 

2.  How  many  cubic  feet  in  the  frustum  of  a  cone  whose  altitude 
is  6  ft.  and  the  diameters  of  its  bases  4  ft.  and  3  feet  ? 

Rule. — To  the  sum  of  the  areas  of  both  bases  add  t7ie  square  root 
of  the  product,  and  multiply  this  sum  by  one-third  of  the  altitude. 

3.  How  many  cubic  feet  in  a  piece  of  timber  30  ft.  long,  the 
greater  end  being  15  inches  square,  and  that  of  the  less  12  inches? 

4.  How  many  cubic  feet  in  the  mast  of  a  ship,  its  height  being 
50  ft.,  the  circumference  at  one  end  5  feet  and  at  the  other  3  feet? 


472  MENSURATION. 

THE    SPHERE. 

929.  A  Sphere  is  a  body  bounded  by  a  uniformly  curved  sur- 
face, all  the  points  of  which  are  equally  distant 
from  a  point  Within  called  the  center. 

930.  The  Diameter  of  a  sphere  is  a 
straight  line  passing  through  the  center  of  the 
sphere,  and  terminated  at  both  ends  by  its 
surface. 

931.  The  Radius  of  a  sphere  is  a  straight  line  drawn  from 
the  center  to  any  point  in.  the  surface. 

932.  To  find  the  surface  of  a  sphere. 

1.  Find  the  surface  of  a  sphere  whose  diameter  is  9  in. 
Operation.— 9  in.  x  3.1416  =  28.2744  in.,  circumference. 

28.2744  in.  x  9  =  254.4696  sq.  in.,  surface. 
Rule.— Multiply  the  diameter  by  the  circumference  of  a  great  circle 
of  the  sphere. 

2.  What  is  the  surface  of  a  globe  3  feet  in  diameter  ? 

3.  Find  the  surface  of  a  globe  whose  radius  is  1  foot. 

933.  To  find  the  volume  of  a  sphere. 

1.  Find  the  volume  of  a  sphere  whose  diameter  is  18  inches. 
Operation. — 18  in.  x  3.1416  —  56.5488  in.,  circumference. 

56.5488  in.  x  18  =  1017.8784  sq.  in.,  surface. 
1017.8784  sq.  in.  x  1876=3053.6352  cu.  in.,  volume. 
Rule.— Multiply  the  surface  by  \  of  the  diameter,  or  ^  of  the  radius. 

2.  Find  the  volume  of  a  globe  whose  diameter  is  30  in. 

3.  Find  the  solid  contents  of  a  globe  whose  radius  is  5  yards. 

934.  To  find  the  three  dimensions  of  a  rectangu- 
lar solid,  the  volume  and  the  ratio  of  the  dimensions 
being  given. 

1.  What  are  the  dimensions  of  a  rectangular  solid,  whose  volume 
is  4480  cu.  ft.,  and  its  dimensions  are  to  each  other  as  2,  5,  and  7  ? 


Operation.— y  4480  -=-  (2  x  5  x  7)  =  4 ;  4  ft.  x  2  =  8  ft.,  height  ; 
4  ft.  x  5  =  20  ft.,  width;  4  ft.  x  7  =  28  ft.,  length. 


REVIEW     OF     SOLIDS.  473 

Rule. — I.  Divide  the  volume  by  the  product  of  the  terms  proportioned 
to  the  three  dimensions,  and  extract  the  cube  root  of  the  quotient. 

II.  Multiply  the  root  thus  obtained  by  each  proportional  term ;  the 
products  will  be  the  corresponding  sides. 

2.  What  are  the  dimensions  of  a  rectangular  box  whose  volume 
is  3000  cu.  ft.,  and  its  dimensions  are  to  each  other  as  2,  3,  and  4  ? 

3.  A  pile  of  bricks  in  the  form  of  a  parallelopiped  contains  30720 
cu.  feet,  and  the  length,  breadth,  and  height  are  to  each  other  as  3, 
4,  and  5.     What  are  the  dimensions  of  the  pile  ? 

SIMILAR    SOLIDS. 

935.  Similar  Solids  are  such  as  have  the  same  form,  and 
differ  from  each  other  only  in  volume. 

Principles. — 1.  The  volumes  of  similar  solids  are  to  each  other  as 
the  cubes  of  their  like  dimensions. 

1.  If  the  volume  of  a  cube  3  inches  on  each  side  is  27  cu.  in., 
what  is  the  volume  of  one  7  inches  on  each  side. 

Operation.— 33 :  73 :  :  27  cu.  in. :  x  =  343  cu.  in.,  volume. 

2.  Tlie  like  dimensions  of  similar  solids  are  to  each  other  as  the  cube 
roots  of  their  volumes. 

3.  If  the  diameter  of  a  ball  whose  volume  is  27  cu.  in.  is  3  in., 
what  is  the  volume  of  one  7  inches  on  each  side  ? 

Operation.— ^27  :  ^343  ::  3  :  x  =  7  in.  diameter. 
REVIEW    OF    SOLIDS. 

Jfft  O  BLEMS. 

936.  1.  What  is  the  edge  of  a  cube  whose  entire  surface  is 
1030  sq.  feet,  and  what  is  its  volume  ? 

2.  What  must  be  the  inner  edge  of  a  cubical  bin  to  hold  1250  bu. 
of  wheat  ? 

3.  How  many  gallons  will  a  cistern  hold,  whose  depth  is  7  ft., 
the  bottom  being  a  circle  7  feet  in  diameter  and  the  top  5  feet  in 
diameter  ? 

4.  What  is  the  value  of  a  stick  of  timber  24  ft.  long,  the  larger 
end  being  15  in.  square,  and  the  less  6  in.,  at  28  cents  a  cubic  foot  ? 


474  MENSURATION. 

5.  If  a  cubic  foot  of  iron  were  formed  into  a  bar  \  an  inch  square, 
without  waste,  wbat  would  be  its  length  ? 

6.  If  a  marble  column  10  in.  in  diameter  contains  27  cu.  ft.,  what 
is  the  diameter  of  a  column  of  equal  length  that  contains  81  cu.  ft.? 

7.  How  many  board  feet  in  a  post  11  ft.  long,  9  in.  square  at  the 
bottom,  and  4  in.  square  at  the  top  ? 

8.  The  surface  of  a  sphere  is  the  same  as  that  of  a  cube,  the  edge 
of  which  is  12  in.     Find  the  volume  of  each. 

9.  A  ball  4.5  in.  in  diameter  weighs  18  oz.  Avoir. ;  what  is  the 
weight  of  another  ball  of  the  same  density,  that  is  9  in.  in  diameter  ? 

10.  In  what  time  will  a  pipe  supplying  6  gal.  of  water  a  minute 
fill  a  tank  in  the  form  of  a  hemisphere,  that  is  10  ft.  in  diameter? 

11.  The  diameter  of  a  cistern  is  8  feet ;  what  must  be  its  depth 
to  contain  75  hhd.  of  water? 

12.  How  many  bushels  in   a  heap  of  grain  in  the  form  of  a  cone, 
whose  base  is  8  ft.  in  diameter  and  altitude  4  feet  ? 

GAUGING. 

937 .  Gauging  is  the  process  of  finding  the  capacity  or  volume 
of  casks  and  other  vessels. 

A  cask  is  equivalent  to  a  cylinder  having  the  same 
length  and  a  diameter  equal  to  the  mean  diameter  of  the 
cask. 

To  find  the  mean  diameter  of  a  cask  {nearly), 
Add  to  the  head  diameter  f,  or,  if  the  staves 
are  but  little  curved,  .6,  of  the  difference  between  the  head  and  bung 
diameters. 
To  find  the  volume  of  a  cask  in  gallons, 

Multiply  the  square  of  the  mean  diameter  by  the  length  (both  in 
inches)  and  this  product  by  .0034. 

1.  How  many  gallons  in  a  cask  whose  head  diameter  is  24  inches, 
bung  diameter  30  in.,  and  its  length  34  inches? 


Opebation—  24  +  (30  —  24  x f )  =  28  in.,  mean  diameter. 
282  x  3 J  x  .0034  =  90.63  gal.,  capacity. 

2.  What  is  the  volume  of  a  cask  whose  length  is  40  inches,  the 
diameters  21  and  30  in.,  respectively? 

3.  How  many  gallons  in  a  cask  of  slight  curvature,  3  ft.  C  in.  long, 
the  head  diameter  being  26  in.,  the  bung  diameter  31  in.  ? 


FORMULAS, 


475 


938. 


1.  The  Diameter  < 


2.  The    Circum- 
ference 


3.  The  Area 


x  3.1416 

-^  .3183 
x  .8862 
-j-  1.1284 
x  .8660 
h-  1.1547 
x  .7070 
-r-  1.4142 
x  .3183 
-s-  3.1416 
x  .2821 
-*-  3.5450 
x  .2756 
-j-  3.6276 
x  .2251 
-f-  4.4428 
x  .15915 
■i-  6.28318 
-j-  3.1416 
x  1.2732 
•*-     .7854 


CIRCLES. 

=  the  circumference. 

y  =  the  side  of  an  equal  square. 

\  =  the  side  of  an  inscribed  eqiti 
[  lateral  triangle. 

[■  c=  the  side  of  an  inscribed  square. 
}  =  the  diameter. 

[■  ==  the  side  of  an  equal  square. 

s=  the  side  of  an  inscribed  equi- 
lateral triangle. 

—  the  side  of  an  inscribed  square. 


=  the  radius. 

=  the  square  of  the  radius. 
i  =  the  square  of  the  diameter. 

x  13aSH?o  \  —the  sq're  of  the  circumference, 
■+■     .07958  ) 


939. 

1.  The  Surface 

2.  The  Volume 

3.  The  Diameter 

4.  The  Circumference 


5.  The  Radius 


SPHERES. 

(  Circumference  x  itsdiam. 
_  J  Radius^2  x  12.5664. 
T"  1  Diameter"1  x  3.1416. 
[  Circumference*  x  .3183. 
f  Surface  x  £  t'fa  diameter. 
_  I  Radius*  x  4.1888. 
"""  I  Diameter*  x  .5286. 
(  Circumference*  x  .0169. 

^/ Of  surface  x  .5642. 
fyOfwtiume  x  1.2407. 
\fOfswrfaM  x  1.77255. 
fy  Of  volume  x  3.8978. 
~  x  .2821. 


(  y  Of  surface 
~  \  ty Of  volume  x  .6204 

6.  The  Side  of  Inscribed  Cube  =  {p^te*  x  .5774. 


476 


REVIEW. 


940. 


SYNOPSIS    FOR    REVIEW. 


r  1.  Definition.     2.  Lines. 


5.    Tri- 
angles. 


1.  Defs. 


2.  Prob- 
lems. 


1.  Defs. 


Quad- 
rilat- 
erals. 


7.  Circle. 


Prob- 
lems. 


Angles.     4.  Plane  Figures. 

Triangle.  2.  Eight-angled  Tri.  3.  Hypothenuse. 
4.  Base.  5.  Perpendicular.  6.  Altitude.  7.  Equi- 
lateral Triangle.  8.  Isosceles  Triangle.  9.  Scalene 
Triangle.  10.  Equiangular  Triangle.  11.  Acute- 
angled  Triangle.    12.  Obtuse-angled  Triangle. 

(  Area  of  Triangle.     ~\ 

Either  Dimension. 

To  find  I  Area  of  a  Triangle.  V  Rule. 

The  Hypothenuse. 

I  The  Base  or  Perp.     J 

1.  Quadrilateral.  2.  Parallelogram.  3.  Rectangle. 
4.  Square.  5.  Rhomboid.  6.  Rhombus.  7.  TVa/tf- 
zoitf.    8.  Trapezium.    9.  Altitude. 

898.)-.-,  (  Parallelogram.  » 

899.  V  f  ■{   Trapezoid.  V  Rule. 

900.  )  area        (   Trapezium.        ) 


f  1.  Defs.       1.  Circte.       2.  Diameter.       3.  Radius. 


2.  Prob- 
lems. 


904. 
905. 
906. 
907. 

908. 
909. 


To  find 


Diam.  or  Circum.    Rule,  1,  2. 
^ira*.    Rule,  1,  2. 
Diam.  or  Circ.    Rule,  1,  2,  3. 
&«(&  of  Ins.  Square.    Rule,  1,  2. 
Area  of  Circular  Ring.    Rule. 
L  Mean  Proportional.    Rule. 


Similar  Plane  Figures.       1.  Defs.       2.  Prin.  1,  2,  3, 4,  5. 

r  1.  Solid  or  Body.  2.  Prism.  3.  Altitude.  4.  Par- 
aUdopipedon.  5.  Cwfte.  6.  Cylinder.  7.  /tyra- 
1.  Defs.  \  mid.  S.  Cone.  9.  Altitude  of  Pyramid  or  Cone. 
10.  Slant  Height.     11.  Frustum.     12.   ^m 

I       13.  Diameter.    14.  Radius. 


9.  Solids.  ■ 


2.  Prob- 


918.1 
919. 
925 
926 
ms.  <{  927.  J-r 
928. 
932. 
933. 
934. 


Conv.Surf.  of  Prism  or  Cyl.  Rule. 
Volume         "  "       Rule. 

Conv.Surf.  ofPyr.  or  Cone.  Rule. 


Volume         "  "  Rule. 

Conv.  Surf,  of  Frustum.  Rule. 

Volume         "         u  Rule. 

Surface  of  Sphere.  Rnl e. 

Fotam*  "       "  Rule. 

Dim.  of  Rectang.  Solid.  Rule. 

,  3.  Similar  Solids.       1.  Defs.       2.  Principles,  1,  2. 


10.  Gauging.       1.  Definitions.       2.  Rules. 


^Wt 

\kfj~~ 


ilfll©  ®¥S®BM. 


& 


3771 


Copyright.  1879,  by  Daniel  W.  Fish. 

The  edges  of  this  cube  are  each  1  Me'ter,  or  10  Dec'i-me'ters,  or  100  Cen'ti- 
me'ters,  in  length. 


*      «       6       e      7       a r      •      a 

Scale,  &  of  the  Exact  Size. 


941.  The  Metric  System  of  weights  and  measures  is  based 
upon  the  decimal  notation,  and  is  so  called  because  its  primary  unit 
is  the  Me'ter. 

942.  The  Me'ter  (m.)  is  the  &#S0  of  the  system,  and  is  the 
one  ten-millionth  part  of  the  distance  on  the  earth's  surface  from  the 
equator  to  either  pole,  or  39.37079  inches. 

Me'ter  means  measure ;  and  the  three  principal  units  are  units  of  length, 
capacity  or  volume,  and  weight. 


478  METKIC     SYSTEM. 

943.  The  Multiple  Units,  or  higher  denominations,  are 
named  by  prefixing  to  the  name  of  the  primary  units  the  Greek 
numerals,  Bek'a  (10),  Hek'to  (100),  Kil'o  (1000),  and  Myr'ia  (10000). 

Thus,  1  dek'a-me'ter  {Dm.)  denotes  10  meters  (m.) ;  1  hek'to-me'ter  (Hm.), 
100  me'ters ;  1  kil'o-me'ter  (Km.),  1000  me'ters ;  and  1  myr'ia-me'ter  (Mm.), 
10000  meters. 

1)44.  The  Sub-multiple  Units,  or  lower  denominations, 
are  named  by  prefixing  to  the  names  of  the  primary  units  the  Latin 
ordinals,  Dec'i  (^),  Gen'ti  (r^),  Mil'li  (T^Vo)- 

Thus,  1  dec'i-me'ter  (dm.)  denotes  T\j,  or  .1  of  a  me'ter ;  1  cen'ti-me'ter  (cm.), 
■xhi,  or  .01  of  a  me'ter ;  1  milli-me'ter  (mm.),  toVo,  or  -001  of  a  me'ter. 

Hence,  it  is  apparent  from  the  name  of  a  unit  whether  it  is  greater  or  less  than 
the  standard  unit,  and  also  how  many  times. 

945.  The  Metric  System  being  based  upon  the  decimal  scale,  the 
denominations  correspond  to  the  orders  of  the  Arabic  Notation  ;  and 
hence  are  written  like  United  States  Money,  the  lowest  denomina- 
tion at  the  right.     Thus, 


o         o         a        -~     <£     a         5         o 

7      0       15.638 


e 


e 


3     *     «     q     g       €     I     I 

The  number  is  read,  67015.638  me'ters.  It  may  be  expressed  in 
other  denominations  by  placing  the  decimal  point  at  the  right  of  the 
required  denomination,  and  writing  the  name  or  abbreviation  after 
the  figures. 

Thus,  the  above  may  be  read,  670.15638Hm. ;  or  67.015638  Km. ; 
or  670156.38  dm. ;  or  6701563.8  cm. ;  or  it  may  be  read, 

6  Mm.  7  Km.  0  Hm.  1  Dm.  5  m.  6  dm.  3  cm.  8  mm. 

Write  3672.045  me'ters,  and  read  it  in  the  several  orders  ;  read  it 
in  kil'o-me'ters ;  in  hek'to-me'ters;  in  dek'a-me'ters ;  in  dec'i- 
me'ters ;  in  cen'ti-me'ters. 

The  names  mill,  cent,  dime,  used  in  United  States  Money,  correspond  to 
mil'li,  cent'i,  dec'i,  in  the  Metric  System.  Hence  the  eagle  might  he  called  the 
dek'a-dollar,  since  it  is  10  dollars ;  the  dime,  a  dec'i-dollar,  since  it  is  A  of  a 
dollar,  etc. 


METRIC     SYSTEM. 


4-79 


MEASURES    OF    LENGTH. 


946.  The  Me'ter  is  the  unit  of  length,  and  is  equal  to  39.37  in. 
or,  1.0936  yd.  +  . 


Metric  Denominations.  U.  S.  Value. 

1  Mil'li-me'ter  =  .03937  in. 
10  Mil'li-me'ters,  mm.  ■=  1  Cen'ti-me'ter  =  .3937  in. 
10  Cen'ti-nie'ters,  cm.  =  1  Dec'i-me'ter  =  3  937  in. 
10  Dec'i-me'ters,  dm.  =  1  3Ie'ter  =  39.37  in. 
10  Me'tees,  M.    =  1  Dek'a-me'ter  =  32.809  ft. 

10  Dek'a-me'ters,  Dm.  =  1  Hek'to-me'ter=19.8842  rd. 
10  Hek'to-me'ters,^.  =  1  Kil'o-me'ter  =  .6213  mi. 
10  Kil'o-me'ters,   Km.  =  1  Myr'ia-me'ter=  6.2138  mi. 

Units  of  long  measure  form  a  scale  of  tens; 
hence,  in  writing  numbers  expressing  length,  one 
decimal  place  must  be  allowed  for  each  denomina- 
tion. 

Thus,  9652  mm.  may  be  written  965.2  cm.,  or 
96.52  dm.,  or  9.652  m.,  or  .9652  Dm. 

1.  The  Me'ter  is  used  in  measuring  cloths  and  short  dis- 
tances. 

2.  The  Kil'o-me'ter  is  commonly  used  for  measuring  long 
distances,  and  is  about  jj-  of  a  common  mile. 

3.  The  Cent'i-me'ter  and  Mil'li-me'ter  are  used  by  mechanics 
and  others  for  minute  lengths. 

4.  In  business,  Dec'i-me'ters  are  usually  expressed  in  Cent'i- 
me'ters. 

5.  The  Dek'a-me'ter,  Hek'to-me'ter,  and  Myr'ia-me'ter  are 
seldom  used,  but  their  valnes  are  expressed  as  Kil'o-me'ters. 


EXER  CISES, 


Read  the  following 


3.9  m. 

36  din. 
428  cm. 
6.57  dm. 


346  Dm. 

57.9  Hm. 
479.6  m. 
36.75  mm. 


451  Hm. 
593.7  Km. 
105.6  Dm. 
6000  Km. 


4  in.     1  ft ni 


13.043  Km. 
500.032  m. 
31045.7  cm. 


480  METRIC     SYSTEM. 

Change  the  following  to  me'ters : 

327  Dm.  947  cm.  0.72  Km.  30674  mm. 

28  Hm.  236  dm.  1.73  Hm.  83.062  cm. 

16.8  Km.  43.5  cm.  35.4  Dm.  4000.5  dm. 

1.  Write  6  kilometers  6  dekameters  6  meters  6  decimeters  6  centi- 
meters.    Ans.  6.06666  Km.,  or  60.6666  Hm.,  or  606.666  Dm.,  etc. 

Write  the  following,  expressing  each  in  three  denominations^: 

2.  24379  dm.;    15032036  cm.:    2475064  mm.;    30471  Dm. 

3.  6704  Hm. ;    85  Km.  ;     120000  m. ;    780109  cm.  ;  75  m. 

Similar  examples  should  be  given,  until  the  pupil  is  familiar  with  the  reduc- 
tion of  higher  to  lower,  and  of  lower  to  higher  denominations,  by  changing  the 
place  of  the  decimal  point  and  using  the  proper  abbreviations. 

947.  To  add,  subtract,  multiply,  and  divide 
Metric  Denominations. 

1.  What  is  the  sum  of  314.217  m.,  53.0G2  Hm.,  and  225  cm.  ? 
Operation.    314.217  m.  +  5306.2  m.  +  2.25  m.  =  5622  667  m.,  Ans. 

2.  Find  the  difference  between  4.37  Km.  and  1246  m. 
Operation.    4.37  Km.  —  1.242  Km.  =  3.128  Km.,  Ans. 

•     3.  How  much  cloth  in  8^  pieces,  each  containing  43.65  m.  ? 
Operation.    43.65  m.  x  8.25  =  384.8625  m.,  Ans. 

4.  How  many  garments,  each  containing  3.5  m.,  can  be  made 
from  a  piece  of  cloth  containing  43.75  Dm.  ? 

Operation.    437.5  m.  -*-  3.5  m.  =  125  times;  hence,  125  garments,  Am. 

Rule. — Reduce  the  given  numbers  to  the  same  denominations, 
when  necessary  ;  then  proceed  as  in  the  corresponding  operations  with 
whole  numbers  and  decimals. 


EXERCISES. 

1.  Add  7.6  m.,  36.07  m.,  125.8  m„  and  9.127  m. 

2.  Express  as  meters  and  add  475  dm.,  3241  cm.,  and  725  mm. 

3.  Add  56.07  m.,  1058.2  dm.,  430765  cm.,  6034.58  m.,  and  express 
the  result  in  kilometers. 

4.  From  8.125  Km.  take  3276.4  m.  Ans.  4.8480  Km. 


METKIC     SYSTEM. 


481 


5.  The  distance  around  a  certain  square  is  3.15  Krn.    How  many 
meters  will  a  man  travel  who  walks  around  it  4  times? 

6.  How  many  meters  of  ribbon  will  be  required  to  make  32  badges, 
each  containing  40  centimeters  ?  Ans.  12.8  m. 

7.  What  will  be  its  cost,  at  15  cents  a  meter  ? 

8.  Find  the  difference  between  25.3  Km.  and  425.25  m. 

9.  If  an  engine  runs  36.8  Km.  in  an  hour,  how  far  does  it  run 
between  8  o'clock  and  12  o'clock  ? 

10.  Iu  what  time  will  a  train  run  from  Boston  to  Albany,  at  the 
rate  of  46.55  Km.  per  hour,  the  distance  being  about  325.85  Km.  ? 

11.  From  a  piece  of  cloth  containing  45.75  m.,  a  tailor  cut  5  suits, 
each  containing  7.5  m.     How  much  remained  ? 

12.  A  wheel  is  3.6  m.  around.     How  many  times  will  it  revolve  in 
rolling  a  distance  of  1.08  Km.  ?  Ans.  300. 


MEASURES    OF    SURFACE. 

948.  The  units  of  square  measure  are 
squares,  the  sides  of  which  are  equal  to  a  unit 
of  long  measure.  1 

100  Sq.  MiY\i-me'ters(sq.mm.)  =     1  sq.  cm. 


100  Sq.  Cen'ti-me'ters 
100  Sq.  Dec'i-me'ters 

100  Sq.  Me'ters 

100  Sq.  Dek'a-me'ters 
100  Sq.  Hek'to-me'ters 


=  1  sq.  dm.  • 
_  j  1  sq.  in . 
~  { 1  Centar  (ca.) 

)1  sq.  Dm. 
1  Ar.  (a.) 
_  j  1  sq.  Hm. 
"  { 1  Hektar  {Ha 
—     1  sq.  Km. 


m 


j 


q.c/n.,  Exact  Size. 

=  0.155  sq.  in. 

=    15.5  sq.  in. 

|       j  10.764  sq.ft. 

j       j  1.196  sq.  yd. 

sq  rd. 

acre. 


I       (3.954 

j        (.0247 


2  471  acres. 
.3861  sq.  mi. 


Units  of  square  measure  form  a  scale  of  hundreds;  hence,  in 
writing  numbers  expressing  surface,  two  decimal  places  must  be 
allowed  for  each  denomination. 

Thus,  36  sq.  m.  4  sq.  dm.  27  sq.  cm.  are  written  36.0427  sq.  m.  ; 
and  6  Ha.  5  a.  3  ca.  are  written  6.0503  Ha,,  or  605.03  a.,  etc. 

1.  The  Square  Me'ter  is  the  unit  for  measuring  ordinary  surfaces  of  small 
extent,  as  floors,  ceilings,  etc. 

2.  The  Ar,  or  Square  Dek'a-me'tcr,  is  the  unit  of  land  measure,  and  is  equal 
to  119.6  sq.  yd.,  or  3.854  sq.  rd.,  or  .0247  acre. 


482  METRIC     SYSTEM. 


EXEJt  CIS  ES. 

1.  Read  36145  sq.  m.,  naming  each  denomination. 

Ans.  3  sq.  Hm.  61  sq.  Dm.  45  sq.  m. 

2.  Write  in  one  number  4  of  each  denomination  from  sq.  Hm.  to 
sq.  mm.,  expressed  in  sq.  Hm.  Ans.  4.0404040404  sq.  Hm. 

3.  Express  the  following,  each  in  three  denominations : 
6  sq.  Km.  6  sq.  Hm.  24  sq.  Dm.  5  sq.  m. ; 

16  sq.  Dm.  8  sq.  m.  4  sq.  dm.  15  sq.  cm. 

4.  In  15  sq.  Hm.  how  many  square  meters? 

5.  What  is  the  surface  of  a  floor  12  m.  long  and  7  m.  wide? 

6.  Add  8  times  4  Ha. ,  7  times  9  a.,  and  12  times  14  ca. 

7.  What  is  the  area  of  a  piece  of  land  42  Dm.  long  and  36  Dm. 
wide?  Ans.  1512  sq.  Dm.,  or  15.12  Ha. 

8.  Divide  125000  ca.  into  8  equal  parts. 

9.  How  many  times  is  2.50  sq.  m  contained  in  5  Ha.  ? 

10.  How  many  meters  of  carpeting  0.6  m.  wide  will  cover  a  floor 
8  m.  long  and  5.7  m.  wide?  Ans.  76  m. 

11.  At  15  cents  a  sq.  m.,  what  is  the  cost  of  painting  a  surface 
20.5  m.  long  and  6.8  m.  wide?  Ans.  $20.91. 

12.  A  man  having  5  Ha  8  a.  7  ca.  of  land,  sold  .3  of  it,  at  $25  an 
ar.     What  did  he  receive  for  what  he  sold  ? 


MEASURES    OF   VOLUME. 

949.  The  units  of  cubic  measure  are  cubes, 

the  edges  of  which  are  equal  to  a  unit  of  long 

1  at.  cy)i.)  Jujjccic-t  tSizc* 
measure. 

1000  Cu.  Mil'li-me'ters  (cu.  mm.)    =     1  cu.  cm.       =      .061  cu.  in. 

,„™^     „     ,.-       ,  {leu.  dm.   |       L0353cu.  ft. 

1000  Cu.  Cen'ti-me'ters  =^T.„     /7, >  =<  ,  n~,.»  ,.      , 

/ 1  Li'ter(J.) )       ( l.OoOi  h.  qt. 

-~wv„     ^    ,.       ,*  \lcu.  m.\         35.3165 cu. ft. 

1000Cu.Dec'i-me'terS  »^  ^ev  {s)  \  =\     .2759cord. 

Units  of  cubic  measure  form  a  scale  of  thousands ;  hence,  in 
writing  numbers  expressing  volume,  three  decimal  places  must  be 
allowed  for  each  denomination. 

Thus,  42  cu.  m.  31  cu.  dm.  5  cu.  cm.  are  written  42.031005  cu.  in. 

The  cubic  dec'i^me'ter,  when  used  as  a  unit  of  liquid  or  dry  measure,  is  called 
a  Wter. 


METRIC     SYSTEM.  483 


WOOD    MEASURE. 

1000  Cu.  Dec'i-me'ters  (cu.  dm.)  )  __  ( 1  cu.  m.    \  __  J  .2759  cord. 
10  Dec'i-sters  (da.)  )        (1  Ster,  s.  )      \  35.3165  cu.  ft, 

10  Sters  =  1  Dek  a-ster,  Ds.   =    2.759  cord. 

Units  of  wood  measure  form  a  scale  of  tens  ;  hence,  but  one  deci- 
mal is  required  for  each  denomination. 

Thus,  9  Ds.  4  s.  7  ds.  are  written  94.7  s.  ;    or  9.47  Ds. 

1.  The  Cubic  Me'ter  is  the  unit  for  measuring  ordinary  solids ;  as  excavations, 
embankments,  etc. 

2.  Cubic  Cen'ti-me'ters  and  Mil'li-me'ters  are  used  for  measuring  minute 
bodies. 

3.  The  Cubic  Me'ter  when  used  as  a  unit  of  measure  for  wood  or  stone  is 
called  a  Ster. 

4  The  common  Cord  is  about  the  same  as  a  6  sters,  or  36  dec'i-sters. 

JEXJER  C  ISES. 

1.  Write  30  Ds.  6  s.  8  ds.  Ana.  30.68  Ds. 

2.  Express  in  cu.  m.,  3  cu.  m.  3  cu.  dm.  3  cu.  cm.  3  cu.  mm. 

Ana.  3.003003003  cu.  m. 

3.  Write  and  read  the  following,  each  in  cu.  dm.,  in  cu.  cm.,  and 
in  cu.  mm.  : 

16  cu.  m.  275  cu.  dm.  ;    204  cu.  m.   .016  cu.  dm.   .024  cu.  cm.  ; 
10  cu.  m.    324  cu.  dm    .016  cu.  cm.    3244  cu.  cm. 

4.  Express  in  cu. meters  and  add  :  7  cu. m.,  55  cu  dm.,  12  cu.  m., 
6  cu.  dm.,  13  cu.  cm.,  10532  cu.  cm.  Ana.  19.071547  m. 

5.  From  36  cu.  m.  subtract  8  times  42  cu.  dm.    Ana.  35.664  m. 

6.  How  many  cubic  meters  of  brick  in  a  wall  16  m.  long,  3  m. 
high,  and  8  dm.  thick?  Ana.  38.4  cu.  m. 

7.  How  many  cu.  meters  of  earth  must  be  removed  in  digging  a 
cellar  16.5  m.  long,  8.2  m.  wide.,  and  3.2  m.  deep? 

8.  In  a  pile  of  wood  9.3  m.  long,  2.8  m.  high,  and  1.5  m.  wide, 
how  many  sters  ?  Ana.  39.06  s. 

9.  At  $2.25  a  ster,  what  would  be  the  cost  of  a  pile  of  wood  5.6  m. 
long,  3.4  m.  wide,  and  2.5  m.  high  ? 

10.  If  a  cu.  centimeter  of  silver  is  worth  $.75,  what  is  the  value 
of  a  brick  of  silver  12.4  cm.  long,  3.6  cm.  wide,  and  2.5  cm.  thick? 


484 


METRIC     SYSTEM. 


MEASURES    OF    CAPACITY. 

950.  The  Li'ter  is  the  unit  of  ca- 
pacity, both  of  Liquid  and  of  Dry 
Measures,  and  is  equal  in  volume  to  one 
cu.  dec'i-me'ter,  equal  to  1.0567  qt.Liquid 
Measure,  or  .908  qt.  Dry  Measure. 

lOMil'li-li'ters,  ml.=l  Cen'ti-li'ter  = 

10  Cen'ti-li'ters,  cl.  =lDec'i-li'ter  = 

lODec'i-li'ters,   dl.  — 1  Liter 
IOLi'ters,  L.  =1  Dek'a-li'ter 

10  Dek'a-li'ters,  Dl  =1  Hek'to-li'ter 

10  Hek'to-li'ters,.HZ.=l  Kil'o-li'ter  or  Ster 

10  Kil  o-li'ters,    El. = 1  Myr'ia-li'ter  (ML) 


Dry  M% 

.61  cu.in. 
6.10   "     " 
.908  qt. 
9.081  "     : 
2.837  bu. 
28.37  bu. 
.308cu. 
:283.72  bu. 


_j  28.37  bu.| 
_|1.308cu.yd}" 


Liquid  M. 

=r.338fl'doz. 
=   .845  gi. 
=1.0567  qt. 
=2.64175  gal. 
=26.4175  u 

=264.175  u 

=2641.75  m 


1.  The  Li'ter  is  used  in  measuring  liquids  in  moderate  quantities. 

2.  The  Hek'to-li'ter  is  used  for  measuring  grain,  fruit,  roots,  etc.,  in  large 
quantities,  also  wine  in  casks. 

3.  Instead  of  the  Kil'o-li'ter  and  Mil'li-me'ter,  the  Cubic  Me'ter  and  Cubic 
Cen'ti-me'ter,  which  are  their  equals,  may  he  used. 


EXERCISES. 

1.  Write  5  kiloliters  5  liters  5  deciliters  5  centiliters. 

Am.  5.00555  Kl.,  or  5005.55  1. 

2.  Read,  naming  each  denomination,  the  following  : 

45624  cl.  ;    306721  ml.  ;    76031  dl.  ;    89764  1. 

3.  In  3846  1.  how  many  cl.  ?    How  many  Dl.  ?    Kl.  ?    dl.  ?    ml.  ? 

4.  Find  the  sum  of  175  1.,  25  HI.,  42  cl.,  and  16  dl. 

5.  From  6  times  25  HI.  take  15  times  36  1. 

6.  Divide  5  HI.  of  corn  equally  among  25  persons.       Ans.  20 1. 

7.  From  a  cask  of  wine  containing  2  HI.  of  wine,  125  1.  were 
drawn  out.     How  much  remained  ? 

8.  How  many  HI.  of  wheat  can  be  put  into  a  bin  3  m.  long,  2  m. 
wide,  and  1.5  m.  deep  ?  Ans.  90  HI. 

9.  What  must  be  the  length  of  a  bin  1.5  m.  wide,  1  m.  deep,  to 
contain  7500  liters  of  grain  ?  Ans.  5  m. 


METRIC     SYSTEM.  485 


MEASURES    OF    WEIGHT. 

951.  The  Gram  is  the  unit  of  weight,  and  is  equal  to  the 
weight  of  a  cu.  cen'ti-me'ter  of  distilled  water. 

A  Gram  is  equal  to  15  432  gr.  Troy,  or  .03527  oz.  Avoir. 

10  Mil'li-grarns,        mg.     =  1  Cen'ti-gram     ==         .1543  +  gr.  Tr. 
10  Cen'ti-grams,       eg.       =  1  Dec'i-grani      =       1.5432+  "      " 

lODec'i-grams,         dg.      =  1  Gram  ^"j^^. 

10  Gkams,  g.        =  1  Dek'a-gram     —         .3"";27  +  "      " 

10  Dek'a-grams,       Dg.     =  1  Hek'to-gram    =       3.5274+  "       ' 

<*  u  t  u  a  *  S  KU'o-gram, )         i  2.6792     lb.  Tr. 

10  Hek'to-grams,      Hg.     =  \\         ,?.„    'V—    <AAAjl„     ,,     , 
9  }   or  Kilo   )        I  2.2040  +  lb.  Av. 

10  Kil'o-grams,         Kg.     =  1  Myrta-gram    =     22.046  +   "      *' 

10Myr'ia-grams,^.,or)  ^^  =  .      tt 

100  Kil  os,  ) 

10  Quin'tals,  Q.,  or  [  _     ( Tonneau,     )  _  I  2204.62+  "      " 

1000  Kilos,  K  \~     {       or  ton f     ( 1.1088  +    tons. 

1.  The  Gram  is  used  for  weighing  letters,  gold,  silver,  medicines,  and  all 
small,  or  costly  articles. 

2.  The  KU'o-gram  or  Kil'o  is  the  weight,  of  a  cu.  dm.  of  water,  and  is  the  unit 
of  common  weight  in  trade,  being  a  trifle  less  than  2|  lb.  Avoir. 

3.  The  Ton  is  the  weight,  of  a  cu.  m.  of  water,  and  is  used  for  weighing  very 
heavy  articles,  being  about  294;  lb.  more  than  a  common  ton. 

4.  The  Avoir,  oz.  is  about  28  g. ;  the  pound  is  a  little  less  than  i  a  kilo. 


EXERCISES. 

1.  Read  340642  eg.  in  grams ;  in  hectograms;  in  kilograms. 

2.  Change  16.5  T.  to  kilos  ;  to  grams  ;  to  decigrams. 

3.  If  coffee  is  $.80  a  kilo,  what  will  5  quintals  cost? 

4.  How  many  boxes  containing  1  gram  each,  will  be  required  to 
hold  1  kilo  of  quinine  ?  Arts.  1000. 

5.  If  a  letter  weighs  3.5  g.,  how  many  such  letters  will  weigh 
1.015  Kg.?  Ans.  290. 

6.  A  car  weighing  6.577  T.  contains  125  barrels  of  salt,  each 
weighing  102.15  K.     What  is  the  weight  of  the  car  and  contents  ? 

7.  Find  the  difference  in  the  weight  of  the  car  and  its  contents  ? 


486  METRIC     SYSTEM. 

952.  To  change  the  Metric  to  the  Common  Sys- 
tem. 

1.  In  3.6  Km.,  how  many  feet? 

operation.  Analysts. — The  meter  is 

3.6  Km.  x  1000  =  3600  m.  the  PrinciPal  unit  of  the  taWe ! 

^~  .  ~„™         +  i^^^rv  .  hence,  reduce  the  kilometers 

39.37  in.  x  3600  =  141782  in.  t0  meters     since  there  are 

141732  in.  -r-  12         =  11811  ft.,  Am.      39.37   inches   in    1  meter,  in 

3600  m.  there  are  3600  times 
39.3T  in.,  or  141732  in.  =  11811  ft.    Therefore,  3.6  Km.  are  equal  to  11811  ft. 

Rule. — Reduce  the  metric  number  to  the  denomination  of  the 
principal  unit  of  the  table;  then  multiply  by  the  equivalent,  and 
redaze  the  product  to  the  required  denomination. 


EXERCISES. 

2.  How  many  feet  in  472  centimeters?  Ans.  15.485  ft. 

3.  How  many  cubic  feet  in  2000  sters  ? 

4.  How  many  gallons,  liquid  measure,  in  325  deciliters  ? 

5.  How  many  gallons  in  108.24  liters?      Ans.  28  gal.  2.77  qt. 

6.  How  many  bushels  in  3262  kiloliters  ? 

7.  How  many  acres  in  436  ars  ?  Ans.  10.774  A. 

8.  In  942325  centiliters,  how  many  bushels  ? 

9.  In  456  kilograms,  how  many  pounds  ?       Ans.  1005.024  lb. 

10.  In  42  ars,  how  many  square  rods  ? 

11.  Change  75  5  hektars  to  acres.  Ans.  186.56  A. 

12.  How  many  gallons  in  24^  liters  of  wine? 

13.  How  many  pounds  of  butter  in  124  kilos? 

14.  In  28  sters,  how  many  cords?  Ans.  7.725  C. 

15.  In  72  kilometers,  how  many  miles  ? 

16.  Change  148  grams  to  ounces  Avoirdupois.       Ans.  5.22  oz. 

17.  Change  150.75  kilos  to  pounds. 

18.  How  many  sq.  rods  in  5  a.  85  ca.  ?  Ans.  23.13  sq.  rd. 

19.  What  is  the  weight  of  24  cu.  dm.  148  cu.  cm.  of  silver,  if  a 
cu.  centimeter  weighs  11.4  g.  ?  Ans.  126.69  lb.  Tr. 


METRIC     SYSTEM.  487 

953.  To  change  the  Common  to  the  Metric  Sys- 
tem. 

I.  In  10  lb.  4  oz.  Troy,  how  many  kilograms? 

operation.  Analysis— The  gram, 

10  lb.  4  oz.  —  10.25  lb.  the  principal  unit  of  the 

10.25  lb.   x  5760  =  59040  gr.  taWe'    \9    W»«»ed    in 

Kf\t\At\  ir^oo  ooot:  w  grams ;  hence,  reduce  the 

59040  gr-lo.432gr.=  3825.75  g.  pounds   and    ounces    to 

3825.75  g.  -=-  1000  =  3.82575  Kg.,  AnS.      grains.      Siuco  15.432  gr. 

make  1  gram,  there  are 
as  many  grams  in  59040  gr.  as  15.432  gr.  is  contained  times  in  59040  gr.,  or 
3825.75  g.  And  since  there  are  1000  grams  in  a  kilogram,  dividing  3825.75  g.  by 
1000  g.,  the  quotient  is  3.82575.    Therefore,  there  are  3.82575  Kg.  in  10  lb.  4  oz. 

Rule. — Reduce  the  given  quantity  to  the  denomination  in  which 
the  equivalent  of  the  principal  unit  of  the  metric  table  is  expressed  ; 
divide  by  this  equivalent,  and  reduce  the  quotient  to  the  required 
denomination. 

EXERCISES, 

2.  In  6172.9  lb  av.,  how  many  kilograms  ?   Ans.  2800.009  Kg. 

3.  How  many  ars  in  a  square  mile  ? 

4.  How  many  cu.  decimeters  in  1892  cu.  feet  ? 

5.  In  892  gr.,  how  many  grams?  Ans.  57.8  g. 

6.  In  2  mi.  272  rd  5  yd.,  how  many  kilometers?  Ans.  4.59  Km. 

7.  How  many  sters  in  261.4  cu.  feet? 

8.  How  many  liters  in  3  bu.  1  pk.  ?  Ans.  114.5  1. 

9.  How  many  grams  in  6  lb.  Troy  ?    In  6  lb.  Avoir.  ? 
10.  How  many  meters  in  3  mi.  272  rd.  ? 

II.  In  1828  cu.  yd.  how  many  cu.  meters?  Ans.  1397.52  cu.  m. 

12.  In  3588  sq.  yards,  how  many  sq.  meters? 

13.  Bought  454  bu.  of  wheat,  at  $3  a  bushel,  and  sold  the  same 
at  $8.75  per  hektoliter ;  how  many  hektoliters  did  I  sell  ?  Did  I 
gain  or  lose,  and  how  much  ?  Ans.  160  HI. ;  gain,  $38. 

14.  In  13  gal.  3  qt.  2  pt.  3  gi.,  how  many  liters? 

Ans.  53.351.+. 

15.  How  many  sq.  meters  of  plastering  in  a  room  18  ft.  6  in. 
long,  14  ft.  wide,  and  9  ft.  6  in.  high?         Ans.  55.367  sq.  m.  + 


488  METEIC     SYSTEM. 


TEST     PEOBLEMS. 

954.     1.  Find  tlie  weight  of  a  barrel  of  flour  (196  lb.)  in  Kg.  ? 

2.  What  is  the  cost  of  a  carpet  for  a  room  10.5  m.  long,  and  8.4  m. 
wide,  if  the  carpet  is  84  cm.  wide  and  costs  $2.75  a  meter? 

Ana.  $288.75. 

3.  A  farmer  sold  540  HI.  of  wheat,  at  $2  a  bushel,  and  invested 
the  proceeds  in  coal  at  $7  per  ton.     How  many  tons  did  he  buy  ? 

Ana.  294.95  T.  +  . 
4  What  is  the  cost  of  a  building  lot  75  m.  long  and  62  m.  wide, 
at  $40  an  ar?  Ana.  $1860. 

5.  A  bushel  of  wheat  weighs  60  lb.  What  is  the  weight  of  5  HI. 
of  wheat,  in  kilograms  ?  Ana.  886.05  Kg. 

6.  What  will  be  the  cost  of  a  pile  of  wood  15.7  m.  long,  3  m. 
high,  and  7.52  m.  wide,  at  $1.50  a  ster? 

7.  The  new  silver  dollar  weighs  412|  gr.  Troy.  How  many 
grams  does  it  weigh  ?  Ana.  26.73  g. 

8.  How  many  acres  of  land  in  24.6  Km.  of  a  highway,  which  is 
20  m.  wide  ?  Ana.  111.57  A. 

9.  A  bin  is  4.2  m.  long,  2.8  m.  wide,  and  1.5  m.  deep.  What  will 
be  the  cost  of  filling  it  with  charcoal,  at  25  cts.  a  hektoliter? 

10.  A  merchant  bought  300  m.  of  silk  in  Lyons,  at  12.5  francs  a 
meter  ;  he  paid  75  cents  a  yard  for  duty  and  freight,  and  sold  it  in 
New  York  at  $6  a  ;p,rd.     What  was  his  gain?         Ana.  $406.55. 

11.  What  price  per  pound  is  equivalent  to  $2.50  per  Hg.  ? 

12.  If  a  man  buys  5000  g.  of  jewels,  at  35  francs  a  gram,  and  sells 
them  at  $15  a  pennyweight,  what  was  his  gain  or  loss  ? 

13.  If  a  field  produces  40  Hi!  of  oats  to  the  hektar,  how  many 
bushels  is  that  to  the  acre?  Ana.  45.93  bu. 

14.  What  price  per  peck  is  equivalent  to  80  cts.  a  dekaliter  ? 

15.  What  will  be  the  cost  of  excavating  a  cellar  18.3  m.  long, 
10.73  m.  wide,  and  3.4  m.  deep,  at  20  cents  per  ster? 

16.  How  many  pounds  Avoir,  are  there  in  96.4  kilos  of  salt  ? 

17.  How  many  liters  will  a  cistern  hold  that  measures  on  the 
inside  5.5  ft.  long,  4  ft.  6  in.  wide,  and  4  ft.  deep  ?  Ana.  2583.38  1. 


METRIC     SYSTEM.  489 

18.  How  many  meters  of  lining  that  is  60  cm.  wide  will  line 
15  m.  of  silk  that  is  75  cm.  wide !  Ans.  18,75  cm. 

19.  A  lady  bought  40.5  m.  of  silk  in  Paris.  What  would  be  its 
value  in  Boston,  at  $4.75  per  yard  ? 

20.  A  bin  is  4  m.  long,  2.3  m.  wide.  How  deep  must  it  be  to 
contain  40  HI.  of  grain  ?  Ans.  4.347  m.  +. 

21.  How  many  sters  of  wood  can  be  piled  in  a  shed  8.5  m.  long, 
5.8  m.  wide,  and  4.2  m.  high  ?  What  would  be  its  value  at  $3.25  a 
cord  ?  Ans.  207.03  s. ;  $185,665. 

22.  A  dray  is  loaded  with  60  bags  of  grain,  each  bag  holding 
8  Dl.  ;  allowing  75  K.  of  grain  to  the  hectoliter,  what  is  the  weight 
of  the  load  in  metric  tons  ?  Ans.  3.6  T. 

23.  How  many  meters  of  shirting,  at  $.18  per  meter,  must  be 
given  in  exchange  for  250  HI.  of  oats,  at  $1.20  per  hectoliter? 

24.  A  merchant  shipped  to  France  50  barrels  of  sugar,  each  con- 
taining 250  lb.,  paying  $2  per  cwt.  for  transportation.  He  sold  the 
sugar  at  $.34  per  kilogram,  and  invested  the  proceeds  in  broadcloth, 
at  $4  per  meter.    How  many  yards  did  he  purchase  ? 

25.  A  cu.  decimeter  of  copper  weighs  8.8  Kg.  What  is  the  value 
of  a  bar  of  the  same  metal  15  dm.  long,  9.6  cm.  broad,  and  6.4  cm. 
thick,  at  $1.30  a  kilogram?  Ans.  $105.43. 

26.  How  many  bricks,  each  20  cm.  long  and  10  cm.  wide,  will 
pave  a  walk  95.4  m.  long  and  2.1  m.  wide;  and  what  will  they 
cost,  at  $1.75  per  hundred?  Ans.  10017  bricks ;  $175,297. 

27.  What  is  the  value  of  a  pile  of  wood  40  ft.  6  in.  long,  4  ft. 
broad,  and  6  ft.  6  in.  high,  at  $6.50  per  dekastere  ? 

28.  What  will  be  the  cost  of  building  a  wall  96  Dm.  6  m.  8  dm. 
long,  1  m.  6  dm.  thick,  and  2  m.  4  cm.  high,  at  $6.75  a  cu.  meter? 

29.  A  wine  merchant  imported  to  Boston  1000  dekaliters  of  wine, 
at  a  cost  of  $.75  a  liter,  delivered.  At  what  price  per  gallon  must 
he  sell  the  same  to  clear  $2000  on  the  shipment  ?       Ans.  $3,596. 

36.  How  many  gallons  of  water  will  a  cistern  contain  that  is  3  m. 
deep,  2  m.  long,  and  1.5  m.  wide ;  and  what  will  be  its  weight  in 
metric  tons  ?  Ans.  2377.575  gals. ;  9  T. 


490 


METRIC     SYSTEM. 


TABLE    OF    EQUIVALENTS. 

955.  The  equivalents  here  given  agree  with  those  that  have 
been  established  by  Act  of  Congress  for  use  in  legal  proceedings  and 
in  the  interpretation  of  contracts. 


1  inch  ==  2.540  centimeters. 
1  foot  =  3.048  decimeters. 
1  yard  =  0.9144  meters. 
1  rod  =r  0.5029  dekameters. 
1  mile  =  1.6093  kilometers. 
1  sq.  in.  =s  0452  sq.  centimeters. 
1  sq.  ft.  =  9.2903  sq.  decimeters. 
1  sq.  yard  =  0.83G1  sq.  meter. 
1  sq.  rd.  =  25.293  sq.  meters. 
1  acre  =  0.4047  hektar. 
1  sq.  mile  =  2.590  sq.  kilometers. 
1  cu.  in.  =  1G.387  cu.  centimeters. 
1  cu.  ft.  =  28.317  cu.  decimeters. 
1  cu.  yard  =  0.7645  cu.  meter. 
1  cord  =  3.624  sters. 
1  liquid  quart  =  0.9463  liter. 
1  gallon  =  0.3785  dekaliters. 
1  dry  quart  —  1.101  liters. 
1  peck  =  0.881  dekaliter. 
1  bushel  =  3.524  dekaliters. 
1  ounce  av.  ==  28.35  grams. 
1  pound  av.  =  0.4536  kilogram. 
1  T.  (2000  lbs.)  =  0.9072  met.  ton. 
1  grain  Troy  =--  0.0648  gram. 
1  ounce  Troy  =  31.1035  grams. 
1  pound  Troy  =  0.3732  kilogram. 


1  centimeter  =  0.3937  inch. 
1  decimeter  =  0.328  foot. 
1  meter  =  1  0936  yds.  ±s  39.37  in. 
1  dekameter  =  1.9884  rods. 
1  kilometer  =  0.62137  mile. 
1  sq.  centimeter  =  0.1550  sq.  in. 
1  sq.  decimeter  c=  0.1076  sq.  ft. 
1  sq.  meter  =  1.196  sq.  yards. 
1  ar  =  3.954  sq.  rods. 
1  hektar  =  2.471  acres. 
1  sq.  kilometer  =  0.3S61  sq.  mi. 
1  cu.  centimeter  =  0.0310  cu.  in. 
1  cu.  decimeter  =  0.0353  cu.  ft. 
1  cu.  meter  =  1.308  cu.  yards. 
1  ster  =  0.2759  cord. 
1  liter  =  1.0567  liquid  quarts. 
1  dekaliter  =  2.6417  gallons. 
1  liter  =  0.908  dry  quart. 
1  dekaliter  =  1.135  pecks. 
1  hectoliter  =  2.8375  bushels. 
1  gram  =  0.03527  ounce  Av. 
1  kilogram  =  2.2046  pounds  Av. 
1  metric  ton  —  1 .1023  tons. 
1  gram  =  15.432  grains  Troy. 
1  gram  =  0.03215  ounce  Troy. 
1  kilogram  =  2.679  pounds  Troy 


PARTIAL    PAYMENTS.  491 

VERMONT   KTTLE   FOE  PARTIAL  PAYMENTS. 

956.  The  General  Statutes  of  Vermont  provide  the  following 
Rule  for  computing  interest  on  notes,  when  partial  payments  have 
been  made : 

'•  On  all  notes,  bills,  or  other  similar  obligations,  whether  made 
'payable  on  demand  or  at  a  specified  time,  with  interest,  when 
payments  are  made,  such  payments  shall  be  applied :  first,  to  liqui- 
date the  interest  that  has  accrued  at  the  time  of  such  payments ; 
and,  secondly,  to  the  extinguishment  of  the  principal! 

"  On  all  notes,  bills,  or  other  similar  obligations,  whether  made 
payable  on  demand  or  at  a  specified  time,  with  interest  annu- 
ally, the  annual  interests  that  remain  unpaid  shall  be  subject  to 
simple  interest,  from  the  time  they  become  due  to  the  time  of  final 
settlement ;  but  if  in  any  year,  reckoning  from  the  time  such  annual 
interest  began  to  accrue,  payments  have  been  made,  the  amount  of 
such  payments  at  the  end  of  such  year,  with  interest  thereon  from  the 
date  of  payment,  shall  be  applied :  first,  to  liquidate  the  simple  inter- 
est that  has  accrued  upon  the  unpaid  annual  interests  ;  secondly,  to 
liquidate  the  annualinterests  that  have  become  due;  and  thirdly,  to 
the  extinguishment  of  the  principal" 

EXEIt  C  ISES. 

$3458.  Bradford,  Vt.,  Sept.  13,  1869. 

1.  For  value  received,  I  promise  to  pay  B.  W.  Colby  or  order  three 
thousand  four  hundred  and  fifty-eight  dollars,  on  or  before-the  first 
day  of  January ,  187 S,  with  interest.  Samuel  S.  Green. 

Indorsed  as  follows :  Dec.  16,  1870,  $100 ;  May  1,  1871,  $1000 ; 
Jan.  13, 1874,  $85 ;  April  13, 1876,  $450.75. 

What  was  due  Jan.  1,  1878?  Ans.  $3239.90. 

$872.  St.  Johnsburt,  Vt.,  Nov.  22,  1868. 

2.  For  value  received,  I  promise  to  pay  James  Ferguson  or  order 
eight  hundred  and  seventy  two  dollars,  on  demand,  with  interest 
annually.  Sylvanus  E.  Boyle. 

Indorsed  as  follows :  April  4,  1869,  $28  ;  July  10,  1872,  $94.40  ; 
Dec.  10,  1874,  $6.72 ;  Jan.  14,  1877, 
What  was  due  Dec.  28,  1878  ? 


492  PARTIAL    PAYMENTS. 


OPEKATION. 

Int.  on     Yearly 
Int.  Int.        Prin. 

Int.  of  prin.  to  Nov.  22,  1869 $52.32    $872 

Am't  of  1st  payment 29.06 

Bal.  of  unpaid  yearly  int.     .*....  23.26 

Int.  of  prin.  to  Nov.  22,  1872 156.96 

Int.  on  1  year's  int.  3  years $9.42 

Int.  on  bal.  of  unpaid  yearly  int.  3  years  .  4.19       13.61 

Am't  of  2d  payment 96.48 

Bal.  of  unpaid  yearly  int 97.35 

Int.  of  prin.  to  Nov.  22,  1875 156.96 

Int.  on  1  year's  int.  3  years 9.42 

Int.  on  bal.  of  unpaid  yearly  int.  3  years  .     17.52 

"12094    254.31 
Am't  of  3d  payment .      7.10 

Bal.  of  int.  on  int 19.84 

Int  of  prin.  to  Nov.  22, 1877 104.64 

Int.  on  1  year's  int  1  year 3.14 

Int.  on  bal.  of  unpaid  yearly  int.  2  years  .  30.52       53.50      412.45 

1284.45 
Am't  of  4th  payment 416.33 

New  principal 868.12 

Int.  of  new  prin.  to  Dec.  28,  1878 57.30 

Int.  on  1  year's  int.  1  mo.  6  d .31 

Due,  Dec.  28,  1878 $925.73 

Explanation.— We  compute  the  interest  for  one  year  from  the  date  of  the 
note,  as  a  payment  is  made  within  that  year,  and  deduct  the  amount  of  the  pay- 
ment at  the  end  of  the  yeaf  from  the  interest  due.  The  balance  of  interest  bears 
interest  till  Nov.  22, 1872.  The  amount  of  the  payment  at  the  end  of  this  year 
exceeds  the  interest  on  interest  due.  We  therefore  deduct  the  amount  of  the 
payment  from  the  total  interest  due,  and  have  a  balance  of  unpaid  yearly  inter- 
est, $97.35,  which  bears  simple  interest  till  Nov.  22,  1875.  At  this  date  the 
amount  of  the  payment  is  less  than  the  interest  on  interest  due.  We  there- 
fore deduct  the  amount  of  the  payment  from  the  amount  of  interest  on  interest, 
and  have  a  remainder  of  $19.84,  which  is  without  interest.  The  amount  of  un- 
paid yearly  interest  at  this  date  bears  simple  interest  till  the  next  balance. 


PARTIAL    PAYMENTS.  493 

The  amount  of  the  fourth  payment,  Nov.  22, 1S77,  exceeds  the  total  interest 
due.  We  therefore  deduct  it  from  the  sum  of  the  interest  and  principal.  The 
remainder  forms  a  new  principal,  which  bears  simple  interest  to  the  settlement 
of  the  note,  Dec.  28, 1878,  and  one  year's  interest  on  the  same  bears  interest  from 
Nov.  22, 1878,  to  Dec.  28,  1878,  which  interest,  added  to  the  new  principal,  gives 
the  amount  due  Dec.  28,  1878—  $925.73. 

In  cases  of  annual  interest  with  partial  payments,  like  the  above 
example,  observe  the  following  notes  : 

1.  To  avoid  compounding  interest,  keep  the  principal,  unpaid  yearly  inter- 
ests, and  interest  on  yearly  interest,  in  separate  columns. 

2.  Deduct  the  amount  of  the  payment  or  payments  at  the  end  of  the  year 
from  the  interest  on  the  unpaid  yearly  interest,  when  it  does  not  exceed  this 
interest.  The  remainder  never  draws  interest,  but  is  liquidated  by  the  first  pay- 
ment that  equals  or  exceeds  it. 

3.  Deduct  the  amount  of  the  payment  or  payments  at  the  end  of  the  year 
from  the  sum  of  the  unpaid  yearly  interests  and  the  interest  on  the  unpaid 
yearly  interests,  when  this  amount  exceeds  the  interest  on  the  interest,  but  is 
less  than  such  sum.  The  remainder  is  a  balance  of  unpaid  yearly  interest  which 
draws  simple  interest  until  canceled  by  a  payment. 

4.  Deduct  the  amount  of  the  payment  or  payments  at  the  end  of  the  year 
from  the  sum  of  the  total  interest  due  and  the  principal,  when  it,  exceeds  the 
total  interest  due.  The  remainder  forms  a  new  principal,  with  which  proceed 
as  with  the  original  principal. 

$5000.  Newport,  Vt.,  Oct.  19,  1802. 

3.  For  value  received,  we  jointly  and  severally  promise  to  pay  John 
Smith  or  hearer  five  thousand  dollars,  sixteen  years  after  date,  with 
interest  annually.  Geo.  S.  Leazer. 

E.  D.  Crawford. 

Indorsed  as  follows:  Jan.  13,  1866,  $393 ;  Sept.  24,  1866,  $48; 
July  10,  1869,  $493.47;  Oct.  14,  1873,  $100;  Dec.  12,  1877,  $3200; 
April  15, 1878,  $65. 

What  was  due  Oct.  19,  1878?    Ans.  $7056.17. 

$420.  Burlington,  Vt.,  March  23,  1872. 

4.  For  value  received,  I  promise  to  pay  Jos.  B.  Vinton  or  order 
four  hundred  and  twenty  dollars,  six  years  from  date,  with  interest 
annually.  Geo.  A.  Bancroft. 

Indorsed  as  follows ;  Oct.  3,  1873,  $40.23 ;  March  1,  1874,  $8 ; 
Sept.  13, 1875,  $33.38. 
What  was  due  March  23, 1878  ?    Ans.  $494.62. 


494 


PARTIAL    PAYMENTS 


$639.  Barton,  Vt.  Aug.  20,  1872. 

5.  For  value  received,  I  promise  to  pay  E.  J.  Baxter  or  order  six 
hundred  and  thirty-nine  dollars,  on  demand,  with  interest  annually. 

Samuel  Macomber. 

Indorsed  as  follows  :  Oct.  14,  1877,  $10  ;  Dec.  24,  1878,  $20. 

What  was  due  March  30,  1879  ?    Ans.  $904.58. 


TABLE. 

Showing  amount  of  $1.00  from  1  to  SO  years,  at  4,  5,  6,  7  and  S  per 
cent.,  Annual  Interest. 


Years. 

4  per  cent. 

5  per  cent. 

6  per  cent, 

7  per  cent. 

8  per  cent. 

Years. 

1     . 

$1  0400 

$1.0500 

$1.0600 

$1.0700 

$1.0800 

.     1 

2    . 

1.0318 

1.1025 

1.123G 

1.1449 

1.1664 

.     2 

o 

1.1248 

1.1575 

1.1908 

1.2247 

1.2592 

.     3 

4    . 

1.1696 

1.2150 

1.2616 

1.3094 

1,3584 

.     4 

5    . 

1.2160 

1.2750 

1.3360 

1.3990 

1.4640 

.     5 

6    . 

1.2640 

1.3375 

14140 

1.4935 

1.5760 

.     6 

7    . 

13136 

1.4025 

1.4956 

1.5929 

1.6944 

.    7 

8    . 

1.3648 

1.4700 

1.5808 

1.6972 

1.8192 

.     8 

9    . 

1.4176 

1.5400 

1.6696 

1.8064 

1.9504 

.     9 

10  . 

1.4720 

1.6125 

1.7620 

1  9205 

2.0880 

.  10 

11  . 

1.5?80 

1.6875 

1.8580 

2.0395 

2.2320 

.  11 

12  . 

1.5856 

1.7650 

1.9576 

2.1634 

2.3824 

.  12 

13  . 

1.6448 

1.8450 

2.0608 

2.2922 

2  5392 

.  13 

14  . 

1.7056 

1.9275 

2.1676 

2.4259 

2.7024 

.  14 

15  . 

1.7680 

1.0125 

2.2780 

2.5645 

2.8720 

.  15 

16  . 

1.8320 

2.1000 

2.3920 

2.7080 

3.0480 

.  16 

17  . 

1.8976 

2.1900 

2.5096 

2.8564 

3.2304 

.  17 

18  . 

1.9648 

2.2825 

26308 

3.0097 

3.4192 

.  18 

19  . 

2.0336 

2.3775 

2.7556 

3.1679 

36144 

.  19 

20  . 

2.1040 

2.4750 

2.8840 

3.3100 

3.8160 

20 

ASSESSMENT    OF   TAXES.  495 


VERMONT  METHOD   OF   ASSESSING  TAXES. 

957.  The  Grand  List  is  the  base  on  which  all  taxes  are  assessed  ; 
it  is  \%  of  the  appraised  value  of  the  real  estate  and  personal 
property,  together  with  the  poll  list. 

The  Poll  List  is  $2.00  for  every  male  inhabitant,  from  21  to  TO 
years  of  age,  except  such  as  are  specially  exempt  by  law. 

The  General  Statutes  of  Vermont  provide  that  the  listers  in  each 
town  shall  make  a  list  of  all  the  real  estate  and  personal  property, 
and  the  number  of  taxable  polls  in  such  town,  and  that  the  said 
list  shall  contain  the  following  particulars  : 

"  First.  The  name  of  each  taxable  person. 

"  Second.  The  number  of  polls  and  the  amount  at  which  the  same  are  set  in 
the  list. 

M  Third.  The  quantity  of  real  estate  owned  or  occupied  by  such  person. 
"  Fourth.  The  value  of  such  real  estate. 

"Fifth.  In  the  fifth  column  the  full  value  of  all  taxable  personal  estate  owned 
by  such  person. 

"  Sixth.  In  the  sixth  column  shall  be  set  the  one  per  centum  on  the  value  of 
all  personal  and  real  estate,  together  with  the  amount  of  the  polls,  which  sum 
shall  be  the  amount  on  which  all  taxes  shall  be  made  or  assessed. 

The  Statu  and  County  Taxes  are  assessed  by  the  Legislature. 

The  minimum  of  the  State  School  and  Highway  Taxes  is  fixed  by 
law,  and  a  higher  rate  left  optional  with  the  town. 

A  Town  Tax  is  assessed  by  vote  of  the  town,  a  Village  Tax  by 
vote  of  the  village,  and  a  School  District  Tax  by  vote  of  the  district. 

EXE  RC  IS  ES. 

1.  The  town  of  Montpelier  voted  a  town  tax  of  $2.60  on  each 
dollar  of  the  grand  list.  The  appraised  value  of  the  real  estate  was 
$702727,  and  of  the  personal  property  $309987,  and  there  were 
740  taxable  polls.  What  was  the  grand  list  of  the  town?  How 
much  money  was  raised  by  this  vote  ?  What  was  John  Hammond's 
town  tax,  who  was  30  years  of  age,  and  whose  property  was  ap- 
praised at  $8927.75? 


496  ASSESSMENT    OF    TAXES. 

OPERATION. 

$702727  + $309987= $1012714,  assessed  value  of  the  property. 
$1012714  x. 01  =$10127.14,  1%  of  the  assessed  value. 
$2.00  x  740= $1480,  the  poll  list. 
$10127.14+ $1480= $11607.14,  the  grand  list. 
$2.60  x  11607.14=  $30178.56,  amount  of  money  raised. 
$8927.75  x.  01 =  $89.28, 1#  of  the  assessed  value  of  John  Ham- 
mond's property. 

$89.28  +  $2.00,  his  poll  list  =  $91.28,  John  Hammond's  grand  list. 
$2.60  x  91 .28 =$237. 33,  John  Hammond's  town  tax. 

2.  The  appraised  value  of  property,  both  real  and  personal,  in 
the  town  of  Rutland,  for  the  year  1878,  was  $3415264.  The  num- 
ber of  taxable  polls  was  2066.  The  town  voted  to  raise  a  tax  of 
$28713.48.     What  was  the  tax  on  a  dollar  of  the  grand  list  ? 

Ans.  $0.75. 

3.  The  appraised  value  of  the  real  estate  in  the  city  of  Burling- 
ton was  $2542373;  of  the  personal  property,  $399937.  There 
were  2040  taxable  polls.  The  city  voted  to  raise  $60305.58  city 
tax.  What  was  the  amount  of  Henry  Cook's  tax,  a  resident,  who 
was  73  years  of  age,  and  whose  real  estate  was  appraised  at  $750, 
and  his  personal  property  at  $475.50  ?  Ans.  $22.06. 

4.  The  grand  list  in  the  town  of  Chelsea  was  $4403.74.  The  ap- 
praised value  of  all  the  property  was  $368774.  How  many  taxable 
polls  were  there  in  that  town  ?  Ans.  358. 

5.  The  estimated  cost  of  schools  in  school  district  No.  8,  in  the 
town  of  Cabot,  for  one  year,  was  $765.  The  amount  of  public 
money  received  from  the  town  was  $71.50.  The  appraised  value  of 
the  real  estate  in  the  district  was  $48545  ;  of  the  personal  estate 
$15428.75 ;  the  number  of  taxable  polls  in  the  district  103.  How 
much  tax  on  a  dollar  of  the  grand  list  must  the  district  vote,  to  pay 
its  expenses  ?  Ans.  $0.82. 

6.  James  Bell  resides  in  Hardwick ;  he  is  44  years  of  age ;  his 
property,  both  real  estate  and  personal,  is  appraised  at  $8975.50. 
Hardwick  voted  a  town  tax  of  $1  60  on  a  dollar  of  the  grand  list. 
The  highway  tax  is  $0.40 :  the  state  tax  is  $0.45  ;  the  state  school 
tax  is  $0.09 ;  the  school  tax  is  $0  86  ;  and  the  county  tax  $0.04,  on 
the  dollar.     What  is  the  amount  of  his  taxes?  Ans.  $315.64. 


MEASURES.  497 


FRENCH   AND   SPANISH  MEASURES. 

958.  The  old  French  Linear,  and  Land  Meas- 
ure* is  still  used  to  some  extent  in  Louisiana,  and  in 
other  French  settlements  in  the  United  States. 


Table. 

12  Lines    =  1  Inch.  6  Feet      =  1  Toise. 

12  Inches  =  1  Foot.  32  Toises  =  1  Arpent. 

900  Square  Toises  =  1  Square  Arpent. 

The  French  Foot  equals  12.8  inches,  American,  nearly. 

The  Arpent  is  the  old  French  name  for  Acre,  and  contains  nearly 
£  of  an  English  acre. 

In  Texas,  New  Mexico,  and  in  other  Spanish  settle- 
ments of  the  United  States,  the  following  denominations 
are  still  used: 

Table. 

1030000  Square  Varas  —  1  Labor     =    177.136  Acres  (American). 
25  Labors  =  1  League  =  4428.4      Acres  " 

The  Spanish  Foot  =  11.11  +  in.  (Am.) ;  1  Vara  =  33£  in.  (Am.) ; 
108  Varas  =  100  Yards,  and  1900.8  Varas  =  1  Mile. 

Other  Denominations  in  Use. 

5000        Varas  Square  —        1  Square  League. 
1000        Varas  Square  =s        1  Labor,  or  -fa  League. 
5645.376  Square  Varas  -  4840  Square  Yards  =      1      Acre. 
23.76    Square  Varas  =        1  Square  Chain  =        ^  Acre. 
1900.8     Varas  Square  =        1  Section  ss  640      Acres. 


TABLE     FOR     INVESTORS. 
959.    The  following  Table  shows  the  rate  per  cent,  of  Annual  Income 
from  Bonds  bearing  5,  6,  7,  or  8  per  cent,  interest,  and  costing 
from  40  to  125. 


Purchase 
Price. 

5%. 

G%. 

11%. 

8fo. 

Purchase 
Price. 

5%. 

6%. 

7fo. 

8?c. 

40 

12.50 

15.00 

17.50 

20.00 

83 

6.02 

7.22 

8.43 

9.63 

41 

12.20 

14.64 

17.08 

19.52 

84 

5.95 

7.14 

8.33 

9.52 

43 

11.90 

14.28 

16.66 

19.04 

83 

5.88 

7.05 

8.23 

9.41 

43 

11.63 

13.95 

16.28 

18.61 

86 

581 

6.97 

8.13 

9.30 

44 

11.36 

13.63 

15.90 

18.18 

87 

5.74 

6.89 

8.04 

9.19 

45 

11.11 

13.32 

15.56 

17.78 

88 

5.68 

6.81 

7.94 

9.09 

46 

10.86 

13.04 

15.21 

1739 

89 

5.61 

6.74 

7.86 

8.98 

47 

10.63 

12.77 

14.90 

17.02 

90 

5.55 

6.66 

7.77 

8.88 

48 

10.41 

12.50 

14.53 

16.66 

91 

5.49 

6.59 

7.09 

8.79 

49 

10.20 

12.25 

14.29 

16.33 

92 

5.43 

6.52 

7.60 

8.69 

50 

10.00 

12.00 

14.00 

16.00 

93 

5.37 

6.45 

7.52 

8.60 

51 

9.80 

11.76 

13.72 

15.68 

94 

5.31 

6.38 

7.44 

8.51 

52 

9.61 

11.53 

13.46 

15.38 

95 

5.26 

6.31 

7.36 

8.42 

53 

9.43 

11.32 

13.20 

15.09 

96 

5.20 

6.25 

7.29 

8.33 

54 

9.25 

11.11 

12.96 

14.81 

97 

5.15 

6.18 

7.21 

8.24 

55 

9.0;) 

10.90 

12.72 

14.54 

98 

5.10 

6.12 

7.14 

8.16 

56 

8.92 

10.70 

12.50 

14.28 

99 

505 

6.06 

7.07 

8.08 

57 

8.77 

10.52 

12.27 

14.03 

100 

5.00 

6.00 

7.00 

8.00 

58 

8.62 

10.34 

12.03 

13.79 

101 

4.95 

5.94 

6.93 

7.92 

59 

8.47 

10.16 

11.86 

13.55 

103 

4.90 

5.88 

6.86 

7.84 

CO 

8.33 

10.00 

11.66 

13.33 

103 

-4.85 

5.82 

6.79 

7.76 

6t 

8.19 

9.83 

11.47 

13.11 

104 

4.80 

5.76 

6.72 

7.69 

62 

8.06 

9.67 

11.29  12.90 

105 

4.76 

5.71 

6.66 

7.61 

63 

7.93 

9.52 

llll 

12.69 

106 

4.71 

5.66 

6.60 

7.54 

64 

7.81 

9.37 

10.93 

12.50 

107 

4.67 

5.60 

6.54 

7.47 

65 

7.69 

9.23 

10.76 

12.30 

108 

4.62 

5.55 

6.48 

7.40 

66 

7.57 

9.09 

10.60 

12.12 

109 

4.58 

5.50 

6.42 

7.33 

67 

7.46 

8.95 

10.44 

11.94 

110 

4.54 

5.45 

6.36 

7.27 

68 

7.35 

8.82 

10.29 

11.70 

111 

4.50 

5.40 

6.30 

720 

69 

7.24 

8.69 

10.14 

11.53 

112 

4.46 

5.35 

6.25 

7.14 

70 

7.14 

8.57 

10.00 

11.43 

113 

4.42 

5.30 

6.19 

707 

71 

7.04 

8.45 

9.85 

11.26 

114 

4.38 

5.26 

6.14 

7.01 

72 

6.94 

8.33 

9.72 

11.11 

115 

4.35 

5.21 

6.08 

6.95 

73 

6.84 

8.21 

9.58 

10.95 

116 

4.31 

5.17 

6.03 

6  89 

74 

6.75 

8.10 

9.45 

10.80 

117 

4.27 

5.12 

5.98 

0  83 

75 

6.66 

8.00 

9.33 

10.66 

118 

4.23 

5.08 

5.93 

6.77 

76 

6.57 

7.89 

9  21 

10.52 

119 

4.20 

5.04 

5.88 

6.18 

77 

6.49 

7.79 

900 

10.38 

120 

4.16 

5.00 

5.83 

G.06 

78 

6.41 

7.69 

8.97 

10.25 

121 

4.13 

4.95 

5.78 

C.61 

79 

6.32 

7.5!) 

8.86 

10.12 

122 

4.09 

4.91 

5.73 

6.55 

80 

6  25 

7.50 

8.75 

10.00 

123 

4.08 

4.87 

5.69 

6  50 

81 

6.17 

7.40 

8.64 

9.87 

124 

4.03 

4.83 

5.65 

6.45 

82 

6.09 

7.31 

8.53 

9.75 

125 

4.00 

4.80 

5.60 

6.40 

<^CZ^zj$M&^^ 


(answers.} 

The  answers  to  the  introductory  and  more  simple  examples  o! 
many  of  the  articles  have  been  omitted. 


Art.  77. 

2.  $5.78. 
2.  $39.18. 
$.  $137.87. 

4.  $247.78. 

5.  $38.58. 
6'.  $27.78. 

7.  $189.75. 

8.  $17.67. 

Art.  79. 

5.  1646. 
1G19. 
$65.94. 
$287.67. 
$376.71. 
4491. 
7504  lb. 
75686. 

10.  72147. 

11.  $696.87. 
25.  $18.12. 
13   $80.87. 
24-  $144.18. 
IS.   105233. 

20.  $220.31. 
27.  181776. 
18   11965. 
25.  $944.66. 
SO.   $719328. 

21.  $3551.05. 
£5.  1547164. 
83.   $0642.23. 
84-   15873173. 
85.   $101550. 

Art.  91. 

10.  $14.11. 

11.  3231. 

12.  $51.24. 


13.   2123  tons. 

U.  2324  ft. 

15.  2324  days. 

26.  $41.23. 

17.  $230.43. 

25.  $202.12. 

ID.  224113. 

£0.  721220. 

#2.  210532. 


88. 


4175. 
3.  151. 
£.  5113. 
5.  $15.21. 
5.  $22.10. 
7.  $25.26. 
?.  2710. 
1.  34213. 
?.  $212.20. 
?.  $6746. 
?.  221533. 

Art.  93. 


8.  1848. 
5.  3883. 

4.  1318. 

5.  4195. 

6'.  2828G  miles. 
7.  26762  acres. 
5.  228670  ft. 

9.  $240.81. 
20.  $95.58. 
11.   $38.03. 
25.  $6.16. 
13.   32358. 
2&  $64.84. 
25  $135.28. 

16.  $157.63. 

17.  8728  rd. 

18.  45736  tons. 
!£5.  12336. 

!  26.   37588. 


57.  69356. 
28.   480J. 
50.  $3323.59. 

30.  $1264.50. 

31.  33798. 
55.  35555. 

33.  $291.35. 

34.  $222.75. 

35.  $3015.05. 

36.  $5524.77. 
87.   10386. 
38.  $695.79. 
50.  $5351.84. 

40.  $101.10. 

41.  474889. 

Art.  95. 

2.  332650. 
5.  $895.66. 

3.  30443. 

4.  6132. 
2517. 
$15.22. 
4190  miles. 
$3640. 
78388  sq.  mi 

10.  3572  ft. 

11.  $53945. 

12.  $9505.67. 

13.  1909609. 
U.   $5044.25. 

15.  $16948.50. 

16.  $1417.16. 

17.  702. 

18.  $36.50. 
20.  8346. 
50.  16552. 

Art.  105. 

12.  $4743. 
25.  $1956. 


24.  $6190. 

50.  $40.50. 

52.  $30.59. 

22.  $622.50. 

23.  $16120. 

Art.  107. 

5.  12771. 
5.  25830. 
4-   34104. 

5.  $1239.30; 
$1713.15. 

6.  $3885.75; 
$4521.60. 

7.  $2209.32; 
$2383.74. 

8.  482400  ; 
430944 ; 
874752. 

0.  2953216  ; 
5606496  ; 
7083104. 
20.  $85692.24; 
$279759.96 . 
$171384.48. 

11.  $2529.25. 

12.  $319192. 

13.  $14064. 

14.  $264958. 

15.  404914. 

16.  186516. 

17.  241768. 

18.  $51188.62. 

20.  17902976. 
80.  $154037.36. 

21.  15704325  da 

22.  2082600  cts. 

23.  1508741097. 

24.  1587862270. 

25.  3654860576. 

26.  8198473608. 


iOO 


ANSWE fi S 


SO 


27.  982275037. 

28.  336<5731415. 
$2715413.50 
$21718.16. 

SI.  416304. 

32.  0. 

33.  947363302. 

34.  5395144320. 

35.  72618. 

36.  $3594.24. 

37.  $4101.25. 

38.  51408 ; 
$7454160. 

39.  277536  ; 
$49956.48. 

Ait.  109. 

2.  $3505.92. 

3.  3605472. 

4.  3906168. 

5.  $19789.44. 

6.  84338.28. 

7.  16810320. 

8.  54793296. 

9.  $109804.80. 

10.  $9212. 

11.  $430.08. 

12.  $19234.32. 

Art.  110. 

5.  $472. 

6.  $1824. 

7.  $840000. 

8.  600000. 

9.  12600000. 

10.  104000000. 

11.  126930871- 

800. 

12.  350310024- 

000. 

13.  96000. 
128000. 
268800. 

14.  $400000. 

Art.  113. 

1.  $1617.30. 
e.   $50.19. 


10 


3.  $829.56. 

4.  $3023.75. 

5.  17920. 

6.  2878. 

7.  37200. 

8.  151218. 

9.  $7198.75. 
$18801, 

Whole. 
$10938, 

Farm. 
$4617, Stock 

11.  $8232. 

12.  $25  loss. 

13.  77050. 

14.  92500. 

15.  $1714.50. 

16.  $43187.32. 

Art.  133. 

15.  1887; 
7303; 
2883. 

16.  47208^ ; 
2754; 
131181*. 

17.  48475|. 
67297|, 
115458|. 

18.  $172.65. 

19.  580H  lb. 

20.  2584|  days. 

21.  $820.50. 

22.  71474f  mi. 

23.  8219  men. 

24.  20116*  A. 

25.  63362  rd. 
1592  bbl. 


26, 


27.  9375  bu. 

28.  $108.50. 

29.  93  oranges. 
SO.  91  yd. 

31. 


Art.  136. 

3.  2340,-,  ; 
2047;:!; 

1424> !!  ; 


2248ft  ; 

2070U; 
1610M; 


Art.  138. 

9.  $9.58. 

10.  $14.89. 

11.  $25.21. 

12.  354  times 

13.  416  " 
2/,.  672       " 

15.  1763     " 

16.  3300  " 
27.  ISflfr  « 
18.  4f|f  " 
22.  642/T  ; 


20.  1083T\V  ; 
25414ft. 

fi.  $1823,%- 
£2.  $97." 
23.  $76. 
#4.  475  acres. 

25.  37  horses. 
$110  left. 

26.  394. 
£7.  5482. 
25.  7198. 
29.  31416. 
5tf.  7071. 
31.  8723. 
£2.  610. 

&?.  28004if|f. 

34.  xmm. 

35.  4321. 

36.  2036. 

J7.  3645^?|. 
55.  7500. 
39.   43785. 
#?.  4629. 
41.   346. 

Art.  139. 

5.  173. 
S.  285. 


4.  4175. 

5.  437. 

7.  1931  ff. 
5.  76671ft. 

9.  4175. 

11.  16401-H. 

Art.  140. 

5.  279JyVj^. 

9.  2824^V*. 

20.  545TW&- 

11.  $43. 
2?.  20  lots. 
13.  84ft. 

Art.  145. 

2.  4920. 
2 .  9  times, 
3   9     " 
4!  394950. 

5.  538. 

6.  443#T. 

7.  $10.78. 

8.  16399. 
2.  $28.15. 

10.  3000  lb. 
22.  $7.50. 

12.  4258a 
25.  718284. 

24.  7  years. 

25.  55552. 
26\  50496. 
17.  7325. 
25.  826776. 
20.  76cts. 
22.  $107. 

22.  $2123~. 

23.  42  weeks. 
#4.  $367. 

25.   $1806. 
£6.  $30247. 
2S.   3823; 

1849. 
20.  $720; 

$530. 
30.   2008  ; 

1781. 


ANSWERS 


501 


31.  $16550; 
$11925. 

32.  24  boxes. 

33.  356  cords. 
$4  cost. 

3L  288. 

35.  2. 

36.  10. 
37K  1476. 
38.  469. 

Art.  165. 

2.  2,  3,  55,  7. 

3.  2\  3,  5,  19. 

4.  3,  5,  163. 

5.  2,  7,  132. 

6.  32,  5,  72. 

7  2,3,5,7,11. 

5.  3,  5,  7,  11. 

5.  2,  32,  163. 

20.  22,  32,  52,  7. 

11.  32,  5,  72. 

12.  11,  31,  41. 

13.  2\  5,  101. 
U.  210,  3,  7. 
i5.  3',  52,  7, 19. 
20.  19,  23,  29. 

17.  2,5,7,11,13. 

18.  8*,  5,  72,  13. 

19.  2,5,7,11,41. 

Art.  170. 

*.  14. 

3.  32. 
4.5. 

5.  18. 

6.  144. 

7.  22. 

8.  42. 
0.  24. 

Art.  171. 

2.  4. 

5.  7. 

4.  27. 

5.  2. 

6.  1. 

7.  13. 


5.  13. 
0.  113. 
20.  17 ;  87. 

11.  124  ;  2. 

12.  12  ft. 
25.  3  bu. 

14.  4329  bags. 
25.  5  ;  9  ;  11  hr 
16.  8162  rails. 

Art.  177. 

*.  2856. 

3.  120. 

4.  450. 

5.  30030. 

6.  13860. 

7.  1680. 
5.  5280. 

Art.  178. 

2.  4896. 
5.  16800. 

4.  51282. 

5.  1560. 

6.  7200. 

7.  3060. 

5.  1680. 
9.  315. 

20.  240. 
11.  180  ft. 
20.  $60. 

13.  384. 
2£  63. 
25.  $4536. 
16.  720  bu. 

Art.  182. 

3.  13. 
4.4. 
J.  14. 

6.  130. 

7.  33. 

8.  61. 

0.  14839. 
20.  16. 
11.  %. 
20.  IX. 
13.  32. 


24-  403. 
25.  U. 
16.  41i. 
27.  10  tons. 
18.  98  bbl. 
20.  8i  tubs. 
20.  16  shillings. 
02.  $.50. 
22.  $.36. 
05.  120  bu. 

24.  44  yd.,  1st. 
22  yd.,  2d. 

25.  $.77. 

26.  144  bu. 
07.  4  chests. 


Art.  209. 

25.  f . 

**  If 

X5.f 


*7.  tVV 

18.  T\. 


00.  in* 

04.   T2&. 

Art.  211. 

5.  Hi4- 

5.  a$*. 

0.  *W*. 

20.  ^p. 

13?  ±1  I1  days. 


15.    115  8JLX 

20.  t^yu. 

Art.  213. 

5.  17£f. 

0.  28|. 

7.  30f. 

5.  18f. 

0.  1018H- 
20.  50..%.     . 
11.  60i. 
20.  98T^V 
52.  1029  Jff, 

2Vrt.  218. 


If. 


Atts 

HI- 
*  tt;B;H> 

&«;!*;«. 

o    a  8  . 
*•  fftf  » 

IS.  |f; 

ft- 

24.  Hi 

tf .  W 
W 

m 

17.  m 


80  »    SIT* 

ft;  If 

If;  it- 
i  tVf; 

If;  If; 


aoo  . 


502 


ANSWERS 


Art.  221. 

3.  lft 
4-  5ft 
5.  83ft 

e.  m&. 

7.  St**. 

5.  9ft 

m  106^. 

11.  6&A. 

12.  490-ft 
JA  251^  yd. 
X*.  9*  yd. 

15.  20141. 

16.  88£. 

17.  191ft 

Art.  224. 
*ft 

r     3  3  1 

*•     8  To' 

6. 
7. 
5. 
5. 
i0.  ft 

u.  ift 

12.  12T»T. 

15.  7ft. 

16.  lJ3ft. 

17.  183ft 
15.  248jb. 
15.  115ft 

Art.  226. 

1.  88&,    the 
greater. 

2.  e&u. 

3.  18%. 

4.  ioi4ft 

5.  T5g. 
6  2  :i 
7.  149^. 

5.  $lff 

&  1^5. 


56rft 

164ft 


10. 

11. 

12.  68ft 

15.  158ft 

14.  328ft 

15.  265-fV 

16.  699^. 

Art.  229. 

2.  ft 

5.  ft 
4-  11A* 

5.  If. 

6.  lft 

7.6|. 

5.  f. 

9.  16f. 

10.  24. 

12.  15. 

15.  2f . 

14.  20. 

15.  126. 

16.  128*. 

17.  255. 
15.  gf. 
10.  72. 
20.  119  J. 
£&  1532. 
23.  1287. 
£4.  5386. 
£5.  39491. 
26.  15099. 
£7.  12756&. 
29.  1212. 

50.  3624. 

51.  7429. 
32.  3729. 
55.  13272. 
34.   10200£. 
55.  23586. 

208993$. 

322. 
36.  $580. 
57.  $15  L 
38.   $1769TV 
50.  $940$. 
40.   $7i  ;  $22i  ; 

|6i);$llft 


Art.  232. 


24.  $11.65$. 

25.  $2293f. 

26.  $7196. 

27.  $5734. 

28.  $47ft 
20.  $if. 

50.  $28i. 

51.  $199ft 
5^.  $73&. 
55.  $11  Off. 
<?4-  424ft 

Art.  235. 

2.  A- 
5.  sfg-. 

4.  AV 

5.  A. 

7.  A- 

0.  6£. 
^0.  16TW 
11.  4^V 
If.  5f&. 

15.   \%3S 


U.  17A. 

15.  $&. 


16.  5ft 

■^    T^7T  A. 

15.  $10ft 
19.  14£. 
^0.  $8ft 
21.  89^. 
«fi  6||  lb. 
23.  176&  lb. 

Art.  238. 

A  117. 
5.  126. 

4.  205f . 

5.  408ft 

6.  877|. 

7.  1486ft 

5.  $147. 

0.  ISA. 

10.  20T8&. 

11.  29|. 

12.  67^. 
15.  $3|. 
14.  6  sons. 

17.  f 
25.1. 
19.  ft. 
^0.  3|. 
^1.  If. 
££.  f . 
j^5.  36. 

**•  *ft 
^5.  If. 
^6.  ££. 
27.  6ft 
^5.  17ft 
29.  ft 

*>•  lift 
^    1B. 

5&   1521&. 

55.  63. 
*4-  8ft 

55   $10450. 

56.  12£  tons. 
37.  $^« 
55.  $2.18$. 

40.  1$. 

41.  1t2^. 


ANSWERS, 


503 


43.  Hi. 

44.  2. 

45.  hi 

46.  6£  mo. 

47.  mi 

48.  m 

49.  If. 

50.  15AV 

51.  m* 

52.  Ifa. 

Art.  242. 

2.  f. 

*•  A- 

5.  f. 

5.  a. 

6.  A. 

A  A- 

5.  TV 

10.  A- 

1*.  h 

13.  h 
U.  H 

15.  x2^. 

16.  A 

Art.  245. 

7       20      48       O 
-£•     ff«    fff    ffj 


2.  84. 
5.  1677H- 
4-  Iftfc 

5.  3043|. 
e.  9072. 
7.  $4612. 
5.  $10588rV 
9.  7X. 
10.  $10946. 

22.  $4. 
12.  $.75. 

23.  $4577^. 


14.  $4101|. 

15.  4|  tons. 
26.  146^  mil 
17.  3f . 

25.  13|  days. 

19.  $192£f. 

20.  $4fY*. 
22.  $5625. 
«*.  Inc'd  A- 
&?.  Dim'd  ^\. 
££.  9||  bu. 


378  bbl. 


47  acres. 

5.0. 

$108. 

30. 

14rv  cords. 

31. 

20  bbl. 

32. 

$1|. 
$1840. 

32. 

94. 

152|  ft. 

35. 
30. 

1    7     .    Ol 
1TTT  >   aV 

$3224  cotton 

$2418  sugar. 

$1488mol'es. 

$9672  total. 

37. 

<WA 

38. 

If. 

3D. 

% 

40. 

iff. 

41. 

m 

43. 

2i. 

4-3. 

22TV 

44. 

AA- 

4&. 

Art.  267. 

18. 

.596. 

10. 

.0625. 

20. 

.0012. 

21. 

.000074. 

M. 

.0000105. 

23. 

.000099010. 

24. 

.437549. 

25. 

.3040012. 

26. 

600.00000- 

024 

27.  495705000. 

00075. 

28.  4735000.- 

00903624. 

Art.  283. 


24. 

A 

25. 

4 

9' 

26. 

27. 

TwAff" 
|l5f. 

28. 

$36£. 

29. 

$9f. 

SO. 

$27f. 

31. 

24fV 

32. 

84TV. 

33. 

38  rV 

34. 

104^- 

Art.  285 

3. 

$.75. 

4.  $.875. 

5. 

.56. 

6. 

.9375. 

7. 

$.8. 

8. 

$.495. 

9. 

.024. 

10. 

.8125. 

11. 

.83333  +  . 

12. 

.25925  +  . 

13. 

.76785  +  . 

14. 

.24666  +  . 

15. 

.60625. 

16. 

.05078  +  . 

17. 

.003125. 

18. 

.005625. 

19. 

.7. 

20. 

.032. 

21. 

$1,875. 

22. 

$.066. 

23. 

101.75. 

225.625. 

11.125. 

8.6625. 

$.934375. 

$4,008. 

12.69. 


Art.  288. 

2.  1.703326. 

599.007. 

$206,874. 

.058815. 

51.180606. 

$275,215. 

150.0660325. 
9.  79.9992. 

10.  111.233  A. 

11.  $70.03. 

12.  1.5547  +  . 

13.  $7062.15. 

14.  387.33  rods. 

15.  $5984.80. 

Art.  290. 

2.  253.86319. 

3.  $533.06. 

4.  $26.6875. 

5.  .376118. 

6.  $161,085. 

7.  1.99655. 

8.  10.040174. 

9.  103.5. 

10.  4.9999875. 

11.  $.25. 

12.  $0625. 

13.  6.3045. 
U.  .238517+. 

15.  1.873125. 

16.  $129.0625. 

17.  .351. 
IS.  .57675. 

19.  .09. 

20.  2194.85  A. 

21.  $6,458  \. 

22.  $411.58. 

23.  1.6625. 

24.  4.1375. 

25.  $95. 

26.  $47.07. 


504 


ANSWERS. 


Art.  293. 


2. 

.33615. 

8. 

14.2162S. 

4. 

.00087. 

5. 

24.5470025. 

6. 

$105,138. 

7. 

$36.0062. 

8. 

572.8. 

9. 

620.7. 

10. 

1.375. 

11. 

676. 

12. 

20.498. 

13. 

.04765625. 

U. 

.0431388. 

15. 

7.03125. 

16. 

15.015. 

17. 

.0084375. 

18. 

1252.6875. 

19. 

$53.5. 

20. 

114.75. 

21. 

.0615. 

22. 

$155.8475. 

23. 

556.718bu.+ 

H. 

$446.25. 

25. 

$438. 

26. 

$14891.925. 

27. 

$53,696  +  . 

28. 

$12300.75. 

29. 

$113,235  +  . 

30. 

$101,175. 

31. 

$389.49. 

82. 

$242,937  +  . 

S3. 

402.788976. 

84. 

4.437. 

35. 

1.69064. 

SB. 

7.03175. 

Art.  296. 

2. 

22.66f. 

3. 

4500. 

4. 

.2. 

5. 

1.25. 

6. 

36.4. 

7. 

4602. 

8. 

73.73 ; 

24.5766+  ; 

5898.4 ; 

85.0730  +  . 

li.  m 


12. 


9.  $95; 
$15,125. 
10.  14?  times ; 
80 
6.42+  " 

I;  6.66|; 
.075. 

3.13133f ; 
313.1331; 
3131.331 ; 
31313.3|; 
313133f. 
387.5 ; 
38.75 ; 
3.875 ; 
.8875. 
.6455. 

50000  times 
$8000. 
.4. 

18.  .16. 

19.  .1344. 

20.  .0175. 

21.  .00734. 

22.  $.72. 

23.  .00001. 
24. 100000. 

25.  121.875. 

26.  .00331 ;  10. 

27.  23  tons. 

28.  12  coats. 

29.  17  horses. 

30.  136  bbl. 

31.  $65,406  +  . 

32.  550  lb. 

33.  $6.25. 
135  lb. 
.831. 
1554. 


13 


14. 
15. 
16. 
11. 


2.07887 +  . 
1744.0598 +  . 


40.  $295. 

Art.  305. 

1.  .4375. 

2.  .8;  .875;  .36; 
.8125;  .575. 
.088;  .385. 


4.  .857142; 

.7;  .81;  .324 

.476190 ; 

.17073. 

6.  .416  ;  .53  ; 

.590;  .36; 

.313. 

7.  .12  ;    125  ; 

2941176470- 

588235 ; 

484375;   .6; 

28125  ;  .088 

238095 ; 

.2288. 

Art.  306. 

2 .  TV 

a      2 

3.   F. 

4>U. 

r    25 

6.  if. 

7.  T%2T- 

8.  #*. 

9.ffi- 

10.  T%\. 

11-  *ft. 

12.  ff . 

13.  if. 

U.  ff. 

15.  W. 

Art.  307. 

^'  TS  >      "2TB" » 

i¥A;  W 

*  7fft;2fM; 

•7    6 

4.  A- 

5.  m- 

6.& 

7.  M. 

S'  jrf*' 


Art.  313. 

4.  $172. 

5.  $19.83-k 
7.  $856. 

8. 


Art.  315. 

2.  $157,875. 

3.  $3986.722. 

4.  $44.83|. 

5.  $4696.30. 

6.  $65,875. 

7.  $438.75. 

8.  $9.1875. 

9.  $40,176. 

10.  $17.71. 

11.  $325.80+. 

12.  $183.15. 
i5.  $212.75. 

14.  $85.93  +  . 

15.  $23.96  +  . 

Art.  316. 

2.  $6.66  +  . 

3.  $30.34  +  . 

4.  $630,70  +  . 

5.  $39.65 

6.  $20,173  +  . 

Art.  320. 

1.  Dr.  $812.72. 

2.  Cr.  $21788.. 

16. 

Art.  327. 

1.  $448.07. 

2.  $1489.46. 

3.  $1489.84. 

4.  $6053.50. 

5.  $81.80. 

6.  $258.85. 

7.  Cr.  Bal., 

$169,675. 

8.  Note  to  Bal., 

$176.16. 


ANSWERS. 


505 


Art.  328. 

1.  $60. 

2.  $59.57. 
5.  $21,375. 

4.  $7.50. 

5.  $3228.34. 

6.  $3.40. 

7.  $.50. 

8.  $1165. 

9.  383.531$. 

10.  $191.10. 

11.  $4.68$. 

12.  $122.50. 

14.  $.15: 

15.  $176,475. 

15.  $104.10. 

17.  23  bu. 

!<?.  $21,125. 

10.  134|  tons. 

25.  21557.47343. 

21.  48  lb.  each. 

22.  $1584,  gain. 
25.  $450. 

24.  $o.25. 

25.  $1.25  per  C. 

26.  .15. 

27.  $4.50. 

28.  $5.03  +  . 
20.  $196.21  +  . 

30.  11001b. 

31.  80  bu. 
52.  1.69 +  . 

33.  $3.40  +  . 

34.  138  bu. 

35.  $74,  cost. 
$59,  selling 

price. 

86.  $232,745  g'n 

Art.  425. 


2.  45515  gr. 
5.  105948  oz. 
^.  910  in. 

5.  68245  min. 

6.  63964  ft. 

7.  2046  in. 

5.  222  eighths. 


9.   43695  sq.ft. 

10.  224800  P. 

11.  8960  A. 

12.  29106  1. 
25.  9896  cu.  ft. 
14.   6216  pt. 
25.  792  qt. 

16.    3  4800. 
27.  1008  gi. 

18.  149181b. 

19.  5480  pwt. 

20.  34785. 

22.  525600  min. 

22.  7948803  sec. 

23.  8784  hr. 

24.  19325. 

25.  200  quires. 
25.  864  doz. 
27.   78330  d. 
25.  2350  ct. 

29.  8280  d. 

30.  $1045.53. 

31.  960  rd. 

32.  563  bbl. 
55.  80  boxes. 
34.  $29.25. 

55.  13440  times. 

36.  1440  min. 

37.  3160  sheets. 

38.  $432. 
55.  2419200. 

40.  11082240. 

41.  414. 96  st.  mi. 

42.  36  of  each. 

43.  876576  hr. 
44'  1485  vols. 

45.  256  pp. 

46.  3625  lb. 

47.  8344  lb. 

48.  2950  lb. 

49.  32620. 
55.  6325  lb. 

51.   31501b.N.Y. 


7545  " 
461824 
1400  lb. 
1800  lb. 
$136,262. 


57.  $124,095  +  , 


58.  $4,825. 

59.  $10.13f. 

Art.  428. 

2.  15  w.  4  da.  9  hr.  40  min. 

5.  10  mi.  8  ch.  20  1. 

4.  2031  lb.  9  oz.  10  pwt 

5.  50  mi. 
5.  1605  A. 

7.  1  sq.  mi. 

8.  125  cu.  ft.  840  cu.  in. 

9.  297  C.  23  cu.  ft. 

10.  15  hhd.  19  gal.  3  qt.  1  pt. 

11.  846  bu. 

12.  2G4  bbl.  26  gal.  3  qt. 

13.  12905  gal. 

14.  Cong.  63,  O.  2,    1 10. 

25.  14  lb.  10  oz.  18  pwt.  22  gr. 

16.  25  T.  15  cwt.  70  lb. 

17.  25  cwt.  37  lb.  15  oz. 

18.  12  lb.  6  oz. 

19.  201  bu. 

20.  VA  bbl. 

21.  203  bu. 

22.  31.72  quin. 

23.  5  w.  1  da.  1  hr.  1  min.  1  sec. 

24.  191  mo.  8  da.  11  hr.  40  min. 

25.  557°  33'  20". 

26.  87  deg.  50  naut.  mi. 

27.  836  gro.  1  doz.  4  pens. 

28.  227*  doz 

29.  251  sc. 

30.  22  Km.  7  Qu.  10  sh. 

31.  151  Buud.  8  Qu. 

32.  411  Cr.  2s. 

33.  2038  fl. 

34.  80  half-sov. 

55.  £44  2s.  2d.  2  far. 

36.  450  fr. 

57.  46sov.  6s.  3.9  +  d. 

38.  200  marks. 

39.  $1689600. 

40.  4725  lb. 

41.  $32.55  N.  Y. 

42.  7°  3'. 

43.  456  da.  12  hr.  45  mis, 
4*.  $1,871. 

45.  $48. 

46.  $56.16. 


506 


ANSWEE8 


Art.  431. 


A  I  gr. 


I  pt. 
.24s. 

&  oz-  • 
I««  rd. 


sq. 

8.  .32  pt. 

9.  If  yd. 
10.  .33  ft. 

|  oz. 

1 1, 

.252  min. 
WW4,-  so.  rd. 


li. 

20. 

13. 

u. 

15. 


Art.  433. 

2.  10s.  lOd. 

3.  §4  3  1  31  gr.  16. 

4.  85  rd.  5  ft.  6  in. 

5.  9  oz. 

6.  3  ft.  9  in. 

7.  8.8  oz. 

8.  17  da.  8§  hr. 
5.  lis.  1.2d. 

m  86  P.  4  sq.  yd. 
5  sq.    ft.    127& 
sq.  in. 

11.  6|  oz.  Avoir. 

12.  14  cu.  ft.   691 1 

cu.  in. 

13.  5°  48'  7.2". 

14.  f  3  5  TT[  36. 

25.  18cwt.961b.  14  oz. 
25.  55  gal.  1  pt. 

17.  16  sq.  yd.  7  sq.  ft. 

36  sq.  in. 

18.  IS  3  1  3lgr.7T\. 
25.  6gro.  10?-  doz. 
20.  2  mi.  101  rd.  6  ft. 

6  16  in.  + 
SI.  3  gal.  3  qt.  1  pt. 
2gi. 

22.  2  pk.  2  qt.  1  pt. 

23.  212  rd. 

24.  4  T.  5  cwt.  55=  lb 

25.  1  A.  60  P. 

26.  2  Cd.  89.6  cu.  ft. 

2 7.  57  rd.  9  ft.  10i  in 


6  Qu.  6  sheets. 
$54.16f. 


ft  gal. 


Art.  435. 

1 

4.  .01  bu. 

5.  .0001  lb. 
5.  .0004  ton. 

.018  ton. 

7.  t/V  cord. 

8.  .00045  oz. 

9.  ¥uW  ton- 

10.  A  da.  ;  .005  da. 

22.  .02  rd. 

12.  H  pt.  less. 
#.  tA*   A. 

Art.  437. 

0.  I  bbl. 
J.  .4375  Cd. 

4.  .22+  hhd. 

5.  Iff  lb. 

6.  A 

7.  .092. 

8.  |X 

5.  .005489  Tp.    • 

25.  .3. 

22.  ff. 

20.  .581  lea. 

13.  A. 

14.  .001625. 

15.  .09. 

26.  |4. 

17.  m, 

Art.  438. 

2.  44352  steps. 

2.  18  h.  45  min. 

3.  $53.1665125. 

4.  $199.25. 

5.  $111.94. 
0.  £60. 

7.  $128.23 J. 

5.  $34,574. 

5.  708  &  bu.,  111. 

657}  bu.,  La. 

634*§  bu.,  N.  Y. 


10.  5714|  bu.;  Ct. 
53331  bu.,  N. 

11.  $332,679. 

12.  15  carats. 

13.  424.98775  A. 

14.  $90. 

15.  $13457.464-. 

16.  $3,525. 
$87,815. 
$2.54+. 

$138.95. 
3125  bu. 
720  centals. 
5  bbl.  152  lb. 


40 '    iTS¥* 

26.  3.02. 

27.  487^  Rm. 
2S.   307sV  Rm. 

Art.  440. 

3.  22  yd.  2  ft.  10  in. 

4.  19  Cd.  3  cd.  ft. 

13  cu.  ft. 

5.  2  lihd.  17  gal. 

2  qt.  3  gi. 

6.  15  h.  28  min. 

7.  60  gal.  1  qt. 

8.  22  cwt.  84  lb. 

14f  oz. 

9.  10  Pch.  9|§  cu.  ft. 
75.  ft,  11  3  2  32gr.5. 
i2.  $133.24. 

20.  $61.50. 

13.   31  yr.  11  mo.  3  da. 

Art.  441. 

3.  25  Cd.  6  cd.  ft. 

4  cu.  ft. 

4.  239  rd.  11  ft. 

5.  8  cwt.  41  lb.  10  oz. 

6.  10s.  7'-d. 

7.  5  lb.  3  oz.  10  pwt. 

8.  1  w.  6  da.  5  hr. 

17  min.  16.8  sec. 
5.  29  gal.  1  qt.  1  gi. 
10.  1  mi.  193.7  rd. 


ANSWERS 


507 


11.  6|  doz. 

12.  |3353  2. 

13.  20  li.  24  min. 

14.  65.44  P. 

15.  8  gal.  3  qt. 

16.  $7306.71  +  . 

17.  13  cwt.  38  lb. 
IS.  30  Cd.    5  cd.  ft. 

14  cu.  ft. 

Art.  442. 

2.  7  yr.  9  mo.  1  da. 

3.  3  yr.  H  mo.  28  da. 

4.  2  yr.  5  mo.  24  da. 

5.  258  da. 

C.  1  yr.  10  mo.  12  da. 
6h. 

8.  204  da. 

9.  157  da.  21  h. 

10.  2  yr.  5  mo.  8  da. 

9  h.  22  min. 

11.  2  yr.  8  mo.  10  da. 

4    lir.    14    min. 
59  sec. 

Art.  443. 

26  bu.  1  pk.  6  qt. 
39  Cd.  3  cd.  ft. 
13  hhd.   42  gal. 

3qt. 
2715  bu. 
Cong.  25  O.  6  3  11 

3  5  111  36. 

7.  £120  18s.  6d. 

8.  189  A.  40  P.  16 

sq.  yd.  6  sq.  ft. 

10.  89  T.  11  cwt.  1  qr. 

19  lb.  14  oz. 

11.  557  yd.  2  ft.  1H  in. 

12.  13  T.  3  cwt.  67.85 

lb. 

13.  $198,796. 
U-  $125.25. 


Art.  444. 

51  A.  31  P.  8  sq.  ft, 
£7  Is.  lid. 
£5  Is.  4f  d. 

£4  8s.  8^d. 


4.  31  bu.  1  pk.  5  qt. 

lpt. 
28  bu.  lpk.  l.Opt. 
23  bu.  2  pk.  2  qt. 

5.  12  yd.  5£f  in. 
6  yd.  2|f  in. 

G.  2  Cd.  5  cd.  ft.  13$ 
cu.  ft. 

7.  17160  rails. 

8.  1  sq.  mi.  42  A.  112 

P.  26  sq.  yd.  8 
sq.  ft. 
0.  337  yd.  1  ft.  7$  in. 

10.  70  times. 

11.  243  boxes. 

13.  9  cwt.  42  lb. 

14.  165  A.  25  P.  24.4 
sq.  yd,,  nearly. 

Art.  448. 

2.  107°  19'  48;]-". 

3.  122°  26'  45 '  W. 

4.  71°  12'  15"  W. 

5.  90th  W. ;  90th  E. ; 

180th  E. 

Art.  450. 

2.  50  min.  21^  sec. 

3.  Hi.    17  min.  24 

sec.  A.  m., or  next 
day. 

4.  5  h'r.  57  min.  49 

sec. 

5.  5  h.  59  min.  51 

sec. 

6.  1  hr.  3  min.  58  sec. 

7.  1  h.  13  min.  32| 

sec. 

8.  51  min.  18  sec. 

9.  1  hr.  40  min.  8  sec. 
10.  6  hr.  28  min.  27 

sec. 
//.  1  h.  33  min.  27 
sec. 

12.  5  hr.  6  min.  15H 

sec.  A.M.  atCinn. 

4  hr.   53  min.   43 

sec.  A.M.  at  Chi. 


4  h.  43  min.  13 
sec.  a.m.  at  St. 
Louis. 

13.  12  h.   7  min.  41 

sec,  at  night,  B. 

4  h.  53  min.  46*- 
sec.  p.m.,  St.  P. 

2  h.  58  min.  6  sec 
p.m.,  Ast.  Or. 

14.  5  h.  46  min.,  later, 

Rome 

5  h.  5  min.  32  sec, 
later,  Paris. 

15.  10  h.  58  min.  37 

sec,  gains. 

Art.  454. 

1.  52  ft.  9'. 

2.  319  ft.  4'  3". 

3.  23  ft.  10'  9". 

Art.  456. 

2.  68  ft. 

3.  55  ft.  10'  3"  2'" 

8'". 

4.  240  ft.  9'  4". 

5.  50  ft,  9'  10"  6'". 

Art.  405. 

1.  63  sq.  yd. 

2.  7i  ft.  wide. 

3.  61  ft.  long. 

4.  152  sq.  yd.  1  sq.ft. 

5.  848  sq.  yd.  4  sq.ft. 

6.  379£  sq.  rd. 

7.  32  sq.  ck.  2  P. 

8.  427£  sq.  ft. 

9.  7  sq.  rd.  1  sq.  yd. 

6  sq.ft.  88 sq. in. 

10.  18  ft.  3  in.  width. 

11.  n  ch.  25  1.  length 

12.  18  yd.  2  ft. 

13.  58TV  sq.  yd. 

14.  48.  planks. 

15.  44  yd. 

16.  260  yd. 

17.  88|  yd. 
is:  81$  yd. 


508 


ANSWERS, 


19.  $98.54  +  . 

20.  $106.48. 
'21.  $81.87. 

22.  $146.40. 

23.  $60.95. 

24.  1080  tiles. 

25.  $608.40. 

26.  $65,475. 

27.  21.29H  squares. 

28.  $139.57. 

29.  $198. 

50.  $27,378. 

51.  840  sods. 

32.  28f  yd. 

33.  13f  rolls. 

34.  $71.60. 

35.  $13,525  +  . 

36.  11316  shingles. 

37.  $25.55. 

33.  $447,989  +  . 

Art.  467. 

1.  90  A. 

2.  32  rd.  wide. 
S.  190*;  farms. 

4.  .625  A. 

5.  264  rd. 

6.  $7594.80  +  . 

7     40 

<?.  25  rd. 
5.  $220  less. 
70.  $4000  gain. 

Art.  468. 

1.  80  A.  ;  4  Sec. 

2.  5760  rails  ; 
$230.40. 

5.  $340  gain. 

4.  240  A. ;  -|-  Sec. 

5.  120  A.  left ; 
$27.20  gain. 

6.  420  A.  left ; 
$635  gain. 

Art.  474. 

1.  96  cu.  ft. 

2.  108  cu.  ft. 

3.  8.V  ft. 

4.  221  cu.  ft. 
6.  208  cu.  yd. 


6.  3  cu.  yd.  26  cu.  ft. 

297  cu.  in. 

7.  7  cu.  yd.  11  cu.  ft, 

200  cu.  in. 

8.  5  cu.  yd.  25  cu.  ft, 

9.  1|  in.,  height. 

10.  8  in.,  height. 

11.  9  ft.  2  in.,  length. 

12.  4840  cu.  ft. 

13.  12ft  Cd. 
U.  8  ft. 

15.  $13,182  +  . 

16.  $166.60. 

17.  8  ft. 

18.  80  cans. 

19.  $410,156  +  . 

20.  24  ft. 

Art.  477. 

2.  31278  ft  bricks. 
5.  60  Pch. 

4.  49ft  Pch. 

5.  62006+  bricks. 

6.  $1607.82  +  . 

7.  $471,663-. 

8.  663  ft  Perches. 

9.  $3276. 
20.  $423.53. 

22.  2214  cu.  ft. ; 
$344.40  cost. 

Art.  481. 

3.  53  J-.      12.  $6.14  + 

4.  93i.      25.  $1,064. 

6.  $4.20.  25.  13. V  ft. 

7.  $15.75.20.  62  J  ft. 
P.  6  ft.      27.  $9.90. 

11.  16  in.    2&  $9,425. 
20.  315  board  ft ; 
26i-  cu-  ft< 

20.  $159,365. 

21.  $90. 

22.  396  posts  ; 
11880  ft.  lumber. 
$274,824,  cost. 

Art.  485. 

2.  96  bu. 
2.  160  cu.  ft. 


5.  15  ft. 

6.  9 1  ft. 

7.  3  ft. 

£.  108$  bu. 
9.  192  bu.  oats ; 
153|  bu.  potatoes. 

10.  57|  bu.  apples  ; 
72  bu.  barley. 

11.  $1920. 

12.  $173.25. 

13.  $205,056/ 
U-  $259,072. 

15.  39;ft  bbl. 

16.  697-i-  tons. 

17.  $861.13  +  . 
IS.  84  tons. 

19.  $27. 

£0.  270  tons. 

$1485. 
21.  $14.75. 

Art.  490. 

2.  149U  gal. 
£  6  bbl. 

£  4041  cu.  ft. 

5.  54?*hhd. 

6.  1968|  lb. 

7.  8500lf  gal. 
5.  5  ft.  7f  in. 

9.  7163.1  cu.  in. 

20.  7.4805+  gal. 

11.  680ft  gal. 
5687i  lb. 

12.  604.8  cu.  in. 

13.  $43.09+  gain. 
U.  m  gal. 

15.  587A  bu. 
20.  23625  lb. 
27.  $22,797  +  . 
IS.  314.1  cu>  a 
2352!!  gal. 
19.  469.39+  Im.Gal. 

Art.  493. 

3.  11  lb.  8  oz.  7  pwt. 

7gr. 

4.  13  1b.  12+  oz. 

5.  $75.46875. 

6.  87i-  oz. 

7.  $57,986  +  . 

8.  52545  gr. 


AHSWEKS. 


509 


Art.  512.        Art.  518. 

g.  243}  lb. 

3.  $6321. 

4.  £263  2s.  6d. 

5.  2912  bu. 
0.  $175. 

7.  $205.49. 

8.  14.076  rd. 

9.  $6014.40. 

10.  $3180.01. 

11.  2  mi.  277  rd. 

51  ft. 
lg.  386 1  ft. 
13  21 1  bu. 

14.  4374  lb. 

15.  123  men. 

16.  Th  yr. 

17.  .004  hnd. 
!<?.  264?  lb. 
19.  $9896.25. 
go.  $677,331  Ex. 

Savings, 
$922.66|. 
gl.  .52  ;  $45760. 
gg.  $3902.40. 


Art.  515. 

2.  25%. 

3.  25%. 
4-  108%. 

5.  5%. 

6.  14f%. 
7.5%. 

8.  54%. 

9.  62i-%. 

10.  73r^%. 

11.  124%. 

12.  7>%. 

13.  75%. 
14-  8%. 
15.  374%. 
iff.  6%. 
17.  1121%. 
!<?.  20%. 
15.  12$%. 
gO.  50%. 
22.  65%. 


5.  $750. 

4.  91.2  A. 

5.  528  lb. 

6.  690. 

7.  5800. 

8.  .6. 
5.  100. 

10.  $750. 
22.  $5450. 
22.  600  bu. 

13.  10080  bbl. 

14.  8000  bu. 

15.  4500  bu. 
25.  $3000. 

17.  $78133.33|. 

18.  $922.25. 

Art.   520. 

g.  2500. 

&  $6000. 

5.  $1250. 

6.  S7400. 

7.  $3392.86. 

8.  36000. 
5.  $2275. 

10.   900  bu. 
22.  800. 
12.   325  A. 
25.  $2480. 

14.  $375.40. 

15.  $31  pr.  A. 
20.  $45  pr.  bale. 

17.  $4398.55. 

18.  $8750. 

19.  $3400,1  st  yr. 
$3570,2d  yr. 

gO.   $208,331. 

Art.  530. 

2.  $349. 

3.  $842.40. 

4.  $636,375. 

5.  $204.86. 
0.  $253.75. 
7.  $1437.60. 


8.  $306.67. 
P.  $144.32. 

10.  $11016. 
22.  $300. 

Art.  531. 

3.  $208,125. 

4.  $11.314,. 

5.  $.17. 
0.  $6.22$-. 
7.  $4.375 ; 

$2.80. 
<?.  $.584. 
$1.06*. 

9.  $.11|  per.lb. 

Art.  532. 

«?.  18f  %  gain. 

4.  \2\%  loss. 

5.  20%. 

6.  28%. 

7.  14f%. 
5.  24%. 
9.   66|%. 

20.  23%. 

11.  50%. 
22.  37-|-%. 


Art.  533. 


3.  $9375. 

4.  $8.80. 

5.  $1  50. 
0.  $14.14. 

7.  $16666.66f, 

5.  A.  $16000  ; 

B.  $10000. 

Art.  534. 

2.  6.86. 

3.  .75. 

4.  $4.91. 

5.  $.20. 

0.  $244,094. 

7.  $183,331 

\9.  $586,661. 

5.  $6553.60. 


Art.  535. 

g.   $1.47. 

3.  $150. 

4.  $1.06|. 


Art.  547. 

$378,125. 

$82.11. 

$379.40. 

$285.19. 

$20.18. 

$584,174. 

$96.90. 


Art.  548. 

2.  8*%. 

3.  5%. 

4.  u%. 

5.  2\%. 
0.  5%. 
7.  6i%. 

Art.  549. 

g.  $2784. 

3.  $3500. 

4.  $9600. 

5.  $9000. 
0.  $960.40. 


Art.  550. 

9.   $3750. 
5.  $583.334,. 

4.  $25372. 


Art.  551. 


$4696.65. 

$3182.55. 

$1500. 

$10618. 

$6400.76 

In  v.; 
$320.04 

Com. 


510 


ANSWERS. 


7.  31000  lb. 

8.  $10623.44. 

9.  $44231.71  Inv. ; 
$1105.79  Com. 

10.   1640  yd. 

Ait.  553. 

1.  48  bu. 

2.  $1700,  lstyr.; 
$1785,  2d  yr. 

3.  24^%. 

4.  $67.50  gain; 
12%  gain. 

5.  $3640. 

6.  $40842  cost. 
$6807  gain. 

7.  $30000. 

s.  m%. 

9.  $468.75. 

10.  Loses  25%. 

11.  25 1  %  nearly. 

12.  5%. 

13.  $4948.125. 

U.  $2964  whole  gain; 
21^  av.  gain  c/Ct 

15.  Prints  @  $.15  ; 
Cassim.@$4.061I; 
Ticking  @  $.25 ; 
Shawls  @  $9.20 ; 
Thread  @  $.875  ; 
Buttons  @  $1  25  ; 
Amt.  @  $729.96. 

16.  $705.12. 

17.  $155.09. 

18.  61788.6  lb. + 

19.  $.50. 

20.  $10582; 
$132  Com. 

21.  §y/e. 

22.  $8,875;  loss4f%+. 

23.  $3040.20  whole 

gain; 
50%  gain  +. 

Art.  567. 

4.  $101.25  int.; 
$551.25  amt.; 
$21  int. ;  $471  amt. 


3.  $71.32  int.; 
$318.32  amt.; 
$16.47  int.  ; 
$263.47  amt. 

4.  $208.33  int. ; 
$708.33  amt. ; 
$22.92  int. ; 
$522.92  amt. 

5.  $3.46  int.  at  6%. 
$4.03  int.  at  7%; 
$4.32  int.  at  7-|-% 

6.  $115.70  at  5%; 
$185.12  int.  at 8%; 
$208.26  int.  at  9%. 

7.  $196.41  int.at 6>-%; 
$235.'/0int.at7i%. 

8.  $58.97  int.  at  10%; 
$73.71  int.at  12^-% 

9.  $888.40  amt. 

10.  $71.37  amt. 

11.  $1176.50  amt. 

12.  $442.50. 


Art.  569. 

2.  $12.58  int.  at  6% 
$8.39  at  4%. 

3.  $92.53(^5%; 
$148.04  @  8%. 

lh  $269.47  @  7%; 

$288.72  @7|. 
5.  $61 .12  int. 
G.  $292.50  int. 

7.  $1204.12  amt. 

8.  $276.52  amt. 
.9.  $41.27  Int. 

10.  $421.99  amt. 

11.  $85.72  Int. 

12.  $13227.50. 


Art.  573. 

2.  $22.70  Int. 

3.  $3.84. 

4.  $38.34. 

5.  $242.94. 

6.  $318. 

7.  $269.34. 


V 


Art.  574. 

$120. 

$.04. 

$10.58. 

$82.36, 

$10.90. 

Art.  575. 

$58.93. 

$8.40. 

$67.67. 

$159,745. 

$67.09. 

$38.11. 

$8.63. 

$3647.61. 

$115.20. 

$1066.36. 

$2010.42. 

$142.45+. 

$18^6.17. 

$13140. 

$291.  > 

S'8  93. 

$£63.83. 

$828.07. 

$1930.60. 

$3025.17. 

$1120.69. 

$76.67. 

$1931.40  loss. 

Art.  577. 

$600,  $792. 

$693(5.09. 

$6069  08. 

$516.71. 

$669.12. 

$334.56. 

$10000. 

Art.  579. 

$1000. 
$1403.08. 
$1500. 
$889.25. 
$650  80. 


ANSWERS. 


511 


Art.  581. 

2.  7%. 

3.  7%. 
4-  8i% 

5.  6%. 

6.  2%  a  month. 

7.  10^%. 

8.  25%;  16f#j 
12^%;  10%. 

9.  100%;  40%; 
28f%;  16|%; 
10%. 

/0.  7i%. 

/2.  The  2d  is  1|£% 
better. 

Art.  583. 

2.  7  mo.  10  d. 
5.  6  yr.  8  mo. 

4.  7  mo.  6  da. 

5.  3  yr.  4  mo.  24  da. 
G.  m  ;  20  ;  16f  ; 

•  13j- ;  10  yr. 

7.  50  ;   40  ;   28}  ; 
25;   16  yr. 

8.  12 1;  61;  25  yr. 

Art.  586. 

2.  $428.76. 

3.  $189.15. 

4.  $1176.14. 

5.  $100.32. 

6.  $41.99  +  . 

7.  $1495.77. 

8.  $53.38. 
10  $1525.64. 

11.  $1540.79. 

12.  $987.23. 

13.  $1934.84. 

14.  $18142.81. 

Art.  589. 

2.  $464.10. 

3.  $7308. 

4.  $11.30. 

5.  $1161.04.. 

6.  $1047.52. 


Art.  597c 

3.  $659.94. 

4.  $30.14. 

5.  $162.25.  f  7$. Of 

Art.  598c 

2.  $312.50. 

3.  $355.16. 

Art.  603. 

2.  $281.83. 

3.  $102.90. 
4-  $1137.61. 

5.  $43.65  in  favor  of 

dis. 

6.  $931.20. 

7.  $838.26. 

8.  45T<&%. 

9.  $931.83. 

10.  $.05  per  bbl.  more 

profitable  to  buy 
at  $8.75  on  6  mo. 

11.  $3677.75. 

Art.  615. 

2.  $6.27  Bk.  dis. 
$591.23  proceeds. 

3.  $1614.48. 

4.  $10839.83. 

5.  Mat.  Oct.  30 ; 

81  days  term  of 
dis.  ; 
$940.38  proceeds. 

6.  Mat  April  8 ; 

46  days  term  of 
dis. ;  * 
$917  proceeds. 

7.  Mat.  Aug.  2  ; 

79  days  term  of 
dis. ; 
$1295.82  proceeds. 

8.  Mat.  Dec.  15  ; 

30  da.  term  of  dis. 
$1281. 77  proceeds. 

Art.  617. 

2.  $1434  20. 

3.  $719.61. 


4.  $1951.03. 

5.  $2291.44. 

6.  $321.46. 

7.  $659.88. 

8.  $368.25. 

Art.  619. 

2.  $188.43  bal.  Julv 

1st. 

3.  $4.90. 

4.  $369.36. 

5.  $327,927. 

Art.  648. 

2.  $34256.25. 

3.  $16856.25.    t  t- a- n  L    1 

4.  $15843.75.  /  $    /  g  °  * 

Art.  649. 

* .  250  shares. 
3.  220      " 

5.  220      " 

G.  480      " 
7.  200      " 

Art.  650. 

2.  $25500. 

3.  $21100. 

4.  $6930. 

Art.  651. 

2.  8|%. 

3.  8'%  bonds  at  110 
f°-%  better. 

4.  6%  bonds  at  84. 
£i%  better. 

5.  9ff%. 

6.  5-H%- 

7.  3H%- 

Art.  652. 


33f%. 
7l|. 

$40. 
75  ;  66|. 


512 


ANSWERS. 


Art.  653. 

2.  $5466.28. 

3.  $338.20. 

4.  $262.60  better  to 

pay  in  currency. 

Art.  654. 

&  $4000 ;  $4035.87  ; 
$4109.59. 

3.  $74000. 

4.  $17.10800. 

6.  Dim.  $26.25. 

6.  $113  per  annum. 

7.  Stock   invest,   is 

$50  better,  or 
f-?%  yearly. 

8.  $21384  in  N.  Y.  S. 

6's; 
$42768  U.  S.  5's 
of  81. 

9.  $792. 

Art.  664. 

2.  $42.75. 

3.  $24.06. 

4.  $187.50. 

5.  $156.25. 

Art.  665. 

2.  \\%. 

3.  \%. 
f.f* 

Art.  666. 

&  $13600. 

3.  $8960. 

5.  $22220.77. 

6.  $40147.91. 

7.  $24500. 

&  $24766.58. 
9.  $9.90. 

Art.  675. 

2.  $284.78. 
5.  $1055.30. 


5.  $527.65. 

6.  $5888.57. 

7.  $4416.57. 

8.  $3263.93. 

9.  $1131.12  loss. 
10.  $7200. 

Art.  685. 

#.  $11350. 

3.  $19072.16. 

4.  $401920. 
7.  $25.09. 
£.  $87.38. 
9.  $112.50. 

10.  $226.50. 

ii.  .0228  tax  rate. 

$214.65. 
12.  $410.95. 
2*.  $224.37. 

14.  $178.13. 

15.  $420000. 

Art.  700. 

2.  $1566.15. 

4.  $4760.51. 

5.  $5153.24. 

6.  $6388.80. 

Art.  701. 


3.  $720. 

4.  $316.45. 

5.  451  shares. 

6.  97  £  %. 

7.  $20108.35. 

Art.  706. 

2.  $2303.25. 

3.  $3317.63. 

5.  $134.78. 

6.  $352.67.    - 

8.  $421.09. 

9.  $566.50. 
11.  $801.94. 
22.  $4621.16. 
25.  $5243.80. 
14.  $3500.40. 


Art.  707. 

2.  £1543  4s.  2d. 

4.  2318.84  marks. 

5.  1664.13  marks. 

7.  31888.83  francs. 

8.  12918.75  francs. 

Art.  711. 

2.  $179.21. 

5.  5.31  francs. 

6.  $4,987. 

7.  £1055  12s.  4d.  ; 
£21  9s.  9.7d. 

8.  $32.78  ind.  ex. 

9.  696.6  guild,  loss. 
10.  $12617.08. 

Art.  726. 

2.  $437.50. 

3.  $1706.25. 

4.  $1843.75. 

5.  $1234.38. 

6.  $63.18. 

7.  $5775. 

8.  $2376.28  duty. 

■■)  /  /  $6815.75  cost  in 
/     currency. 

9.  $1755.89. 
10.  $987.08. 

Art.  733. 

2.  3  mo.  25  da. 

3.  6  mo.  26  da.  time 
of  Cr. ; 

June  27, 77  Eq.  time 

4.  May  5,  1875. 

5.  5  yr.  20  da.  from 
date  of  last  paym't 

7.  Nov.  26,  Eq.  time. 

8.  73  da.  term  of  Cr.; 
Feb.  26,  Eq.  time. 

9.  Mar.  7,  Eq.  time. 
$1178.01     cash 

value. 

Art.  734. 

2.  Aug.    19,    1875, 

Eq.  time. 

3.  June  7.  187a 


ANSWERS. 


513 


4.  June  27,  1874 ; 
Dis.  $149.28. 

5.  Apr.  23,  1874. 

6.  $2837.02. 

7.  May  20,  1875. 

Art.  737. 

2.  Dec.  13,  Eq.  time. 

3.  Dec.  19. 

4.  Jan.  24,  1879. 

Art.  738. 

2.  Mavl8; 
$1486.17  due. 

3.  Dec.  5,  1875. 

4.  $2069.59. 

5.  Oct.  27  ; 
$2102.58. 

6.  $1272.33. 

Art.  739. 

2.  $2331.65  Sales  ; 

$762.83  Charges ; 
$1568.82  Net  pro- 
ceeds ; 
Bal.  due,  Dec.  27. 

3.  $3966.25  Sales  ; 
$412.98  Charges ; 
$3553.27  Net  pro- 
ceeds ; 

Eq.  time  Apr.  14, 
1875. 

Art.  767. 

2.  60  bu. 

3.  $100. 

4.  $4.05. 

6.  44£bbl. 

Art.  770. 

3  9  horses. 
4.  100  yd. 
J.  16  men. 

6.  96  sheep. 

7.  $5355. 

8.  7  hr.  13*-  min. 

9.  355  bu. 


10.  \\2\  mi. 

11.  59 1  da. 

12.  $7320. 

13.  9  yd. 

u.  %n\. 

15.  46  A.  134  P. 

16.  $63. 

17.  $10958.90. 

18.  $3.25. 

19.  $89.60. 

20.  $120. 

21.  2  yr.  6  mo. 

Art.  772. 

2.  43 1  tons. 

3.  5.i-  weeks. 

4.  432  mi. 

5.  15  da. 

Art.  774. 

0.  $498.08. 

4.  1120  bu. 

5.  $6428.57. 

6.  II4/3  ream- 

7.  220  |  Cd. 
S.  $52!79. 
9.  9  men. 

10.  546  bbl. 

11.  2080  lb. 

22.  $100. 

13.  266605f  brick. 

14.  $236.25. 

15.  694?  yd. 
25.  $1728. 
17.  5  da. 
25.  150  yd. 

19.  3  yr.  4  mo.  24  da. 

20.  $1 1.66|. 

21.  9  men. 

22.  8.116  ft. 

23.  $48. 
&£  $53.08. 
25.  1.6  mo.  + 

Art.  782. 

3.  A's  share  $320. 
B's  "  $216. 
C's     "     $184. 


4.  A.  $303.45. 
B.  $337.17. 
C  $404.61. 
D.  $682.77. 

5.  A.  $1710. 
B.  $870.20. 

6.  A.  $6000. 

B.  $8402.25. 

C.  $5055.75. 

D.  $3042. 

7.  $5785.20,  the  first; 
$5142.40,    the 

second. 

8.  $3516.80  A's  gain; 
$5861.333- B's  " 
$8205.86|  C's  " 

9.  $269559.55    Re- 

sources ; 

$26434.55    Lia- 
bilities ; 

$243125  Stock  ; 

$125000    Origi- 
nal capital ; 

$118125  net  gain; 

$56700  Ames' 
share  ; 

$37800  Lyon's 
share ; 

$23625  Clark'a 
share. 


Art.  783. 

2.  $2400  Barr ; 
$2666.66|  Banks ; 
$2033. 33|  Butts. 

3.  $388,704+  A.; 
$249,169+  B.; 
$112,122  C. 

4.  $1344.164  A.; 
$2027.836  B. 

5.  $5700  A.; 
$3760  B. ; 
$1340  C. 

6.  $1688.434 

Crane : 


Childs ; 
$2012.703  Coe. 


514 


ANSWERS. 


Art.  787. 

2.  $.82. 

3.  $.80  per  bushel. 

4.  $6  gain. 

5.  $6.16. 

Art.  788. 

3.  2  lb.  of  first ; 

2  lb.  of  second ; 

3  lb.  of  third. 

4.  1  at  $4 
5  at  $5 
3  at  $( 

1  at  $8. 

5.  3  bbl.  at  $5^  ; 
3  bbl.  at  $6  ; 

2  bbl.  at  $7f. 

6.  3  gal.  at  $1  20  ; 

3  gal.  at  $1.80  ; 

15  gal.  at  $2.30  ; 
8  gal.  water. 

Art.  789. 

2.  10  cows  at  $32  ; 
10  cows  at  $36  ; 
60  cows  at  $48. 

3.  10  lb.  at  $.80  ; 
10  lb.  at  $1.20  ; 
70  lb.  at  $1.80. 

4.  12  yd.  at  $3^;. 

16  yd.  at  $l|. 

5.  150  acres. 

Art.  790. 

2.  30  men,  5  women, 

20  boys. 

3.  831-  gai   water. 

4.  16,  24,  4,  and  12 

da.  respectively. 

Art.  792. 

1.  72  and  48. 

2.  D's  age  16  ; 
E's  age  24 ; 

►      F's  age  84 

3.  15  bu. 

4.  18  da. 
A.  84  da. 


Starch  $2  a  box  ; 
Soap  $3. 
8%  da. ; 

First  in  26 f  da.; 
Second  in  40  da. ; 
Third  in  20  da. ; 
$180  share  of  1st 
$120  share  of  2d  ; 
$240  share  of  3d. 
14  bbl.  at  $10; 
6  bbl.  at  $7. 
16  min.  2iT9r  sec. 

past  3  o'clock. 
Wheat  $1.33*-  per 

bu. ; 
Oats  $.50  per.  bu. 
8  da. 
$347.71. 
50  bu. 

I 

27 %  nearly. 
$7384A  younger : 
$1107611  elder. 
146f  ft. 


$960  first ; 
$720  second ; 
$840  third. 
$1570.31. 
506  lb. 

Oct.  26, 1875. 
$37439.998; 
$33345; 
$27359.999; 
$25106.82. 
$1.60. 
42  geese ; 
58  turkeys. 
$5700. 

$282.24  Sim.  Int. ; 
$2202.24  Amt. ; 
$295.56  Com.  Int. ; 
$2215.56  "  Amt.; 
$1673.93+  Pres- 
ent Worth  ; 
$246.07  True  Dis. ; 
$283.20  Bk.  Dis. ; 
$1(586.80  Proc'ds; 
$2252.199  Face. 


28.  $3:5.79. 
$473  69. 
$710.52. 

29.  $900,  July  28. 

30.  $.97|. 

31.  $10665.80  in  U.  S 

6's,  5-20. 
$21331.60  in  U.S. 
5's  of  '81. 

32.  A.  3600  bu. ; 

B.  1200  bu.'; 

C.  1200  bu. 

33.  $1.72. 

34.  A. 

35.  $5614.27  Net 

Proceeds. 
July  10,  Eq.  time. 

36.  $6400  M.'s  Cap. ; 
15  mo.  N.'s  time. 

37.  $2023.22;  Apr.  24. 

38.  $2244.66. 

Art.  802. 

2.  1369;  1764; 
3136;  5625. 

3.  3375  ;  5832  ; 
74088;  157464. 

4.  3969  ;  110592  ; 
1048576  ;  248832. 

r.        4  0.       1728 

°-  2  5tf  >  Wffr 
7-  ffih ;  44*- 

8.  645.16. 

9.  1191016. 
10.  1958TV 

11       1  4  (!  1  1  \ 

12.  .00116964. 

13.  .015625. 

14.  46733.803208.  ■ 

15.  .065528814274496, 

16.  33169;: :». 

17.  16.6056  j, 

18.  24.76099. 

19.  .000000250047. 

20.  1520875. 

21.  2023. ?.\. 

22.  5.887". 

23.  640000. 

24.  2540.0390625. 
25-  125.      26.  1200. 


ANSWERS. 


515 


Art.  803. 

5.  1764. 

4.  2304. 

5.  3136. 
0.  9604. 

7.  15625. 

8.  11025. 

9.  50625. 

10.  38809. 

11.  116964. 

Art.  804. 

■»  39304. 

4.  110592. 

5.  262144. 
0.  857375. 
7.  1953125. 

Art.  810. 

*.  8  ;  16  ;  24  ; 

81. 
5.  9  ;  14  ;  21 ; 

15. 

Art.  813. 

2.  85. 

4.  242. 

5.  98. 
0.  115. 
7.  109. 
5.  997. 
9.  1432. 

10.  5464. 
12.  |f. 

#.  & 

15.  .035. 

16.  14.0048  +  . 

17.  1.5005 +  . 

18.  7.625. 

19.  4.213  +. 

20.  103.9. 

21.  59049. 

22.  3.00001654- 

23.  5.6568544. 

24.  1.5411. 

25.  .912874. 

26.  .04419. 


27.  36.37. 

8&  1.507484. 

29.  64. 

50.  tV 

51.  1. 
32.  1.78  4-. 
55.  72. 
34.  90. 
55.  480.8827. 

Art.  815. 

1.  1008  ft. 

2.  240.33  rd. 
5.  52  rd. 

4.  200.56  rd. 

5.  145|  rd. 
0.  $187.20. 

Art.  819. 

5.  25. 

4.  55. 

5.  101. 
0.  165. 

7.  1015. 

8.  1598. 
10.  g. 

X*r  ft 

12.  1.42  4- . 

15.  34. 

14.  .45. 

15.  2.34. 
10.  4624. 
17.  .0809. 
15.  .7936. 
19.  5.73+. 

m  M- 

£/.  .5569. 

22.  1. 
25.  14.75. 
24.  60.8. 

Art.  821. 

1.  3  ft. 

2.  8  ft. 
5.  2  ft. 

4.  12150  sq.  ft. 

5.  5  ft.  84  in. 

6.  9  ft.  5.3  4  in. 

7.  8  ft.  1.4  in. 


Art.  822. 

2.  274. 

5.  32. 

4.  543. 

5.  1.05 +  . 

Art.  829. 

5.  8.       6.  149 


4.  17. 

5.  33. 


7.  16. 

5.  m 


Art.  830. 

fc  2.        5.  4. 
5.  2.       0.  7% 
£  f,        7.  3V 

Art.  831. 

2.  9. 
5.  15. 
4.4 

5.  27. 
0.  11  yr. 

Art.  832. 
&  600. 
5.  154. 

4.  125000. 

5.  78. 

6.  57900  ft. 

Art.  840. 

.    5.   Tjfc. 

I   4-  6144 
5.  3. 
0.  $524288. 

7.  $315,619  +  . 
5.  $10485.76. 

Art.  841. 

2.  i.       4-  5. 
5.  5.       5.  3. 

842. 


Art.  843. 

5.  765. 

i  IF 

0.  li 
7.  2. 
5.  280. 
9.  $1023. 
10.  $5314.40. 

Art.  853. 

5.  $3819.75. 

4.  $1292.31. 

5.  $3625. 
0.  6  yr. 
7.  7%. 

5.  $375.30. 

Art.  854. 

5.  $300. 

4.  $3907.665  +  , 

5.  $1182.05  +  . 

6.  $3725.87  +  . 

7.  $629,426  +  . 


Art.  882. 

2.  600  sq.  ft. 
5.  42-j3^  sq.  ft. 

4.  22  A.  6  sq. 

ch.  13.45  P. 

5.  $449.07. 
0.  $147. 

7.  210  sq.  ft. 

Art.  883. 

2.  4|  ft. 
5.  13  in. 

4.  28  rd. 

5.  672  rd. 

yd. 

0.  8*  ch. 
7.  50  rd. 


Art. 

2.  9. 
5.  7. 
4.8. 


«* 


Art.  884. 

p.  111.85  sq.  ft 

1  3  sq.ft.  1.7 

sq.  in. 

4.  13  A.  41.76 

P. 

5.  349.07  sq.  ft 


516 


ANSWERS 


Art.  886. 

2.  39  ft. 

S.  25  ft.  7.34  in. 

4.  33.97  ch. 

5.  28  ft.  3.36  in. 

Art.  887. 

2.  45  yd. 

3.  19  ft.  2.5  in. 

4.  360  ft.  6f  in. 

5.  20  ft. 

Art.  898. 

2.  84  sq.  ft. 
8.  6%  A. 

Art.  899. 

2.  11178  sq.  ft. 

3.  28|  sq.  ft. 

4.  2  A. 

Art.  900. 

2.  213  sq.  ft. 

8.   17  A.  8  ch.  3.4  P. 

Art.  904. 

3.  15  ft.  10.08  in. 
4-  5  ft.  10.67  in. 

5.  5  ft. 

6.  7  ft.  3.96  in. 

Art.  905. 

4.  318.3  A.  + 

5.  114.59  A. 

Art.  906. 

3.  7  rd. 

4.  19.098  ft.  Diam. 
59.998  ft.  Circum. 

Art.  907. 

2.  141.42  ft. 

3.  23.4  yd.  + 

4-  7.07  ft.  + 

Art.  908. 

2.  32.98  sq.  ft.  + 

3.  796.39  sq.  ft. 

4.  1  A.  75.62  P.  land. 
78.54  P.  water. 


Art.  909. 

2,  84. 
3-  28. 

4.  A- 

5.  32  lb.  13.7  oz. 


Art.  910. 

369  rd.  L.; 

123  rd.  W.      ^ 

3.5  in. 

221;    238;    and' 
255  ft. 


126.78  rd. 

Art.  911. 

$185.53. 
35.35  ft.  + 
403.7  rd.  + 
$5812.50. 
$32.40. 
28.66  P.  + 

5  A. ;  or  twice  as 
large. 

$724.75. 

20  ft. 

98  A.  28  P. 

14.645  ft. 

294 rd.;  45.36 rd. 

14  A.  150.4  P. 

6  in. 

Art.  918. 

207.34  sq.  ft. 
168|-  sq.  ft. 
263.89  sq.  ft. 
301.177  sq.  ft. 

Art.  919. 

274|  cu.  ft. 

$27. 

73.63  cu.  ft. 

$53.70. 

Art.  925. 

824  67  sq.  ft. 
429]  sq.  ft. 
512.9  sq.  ft. 
$25. 


Art.  926. 

3.  39.27  cu.  ft. 

4.  $29.23. 

5.  192000  cu.  ft.  voL 
22284.6  sq.  ft. 

surface. 

Art.  927. 

2.  345  sq.  ft. 

3.  256f  sq.  yd. 

Art.  928. 

2.  58.1196  cu.  ft. 

3.  38  \  cu.  ft. 

4.  64.99  cu.  ft. 

Art.  932. 

2.   28.27  sq.  ft. 
12.57  sq.  ft. 

Art.  933. 

2.  8cu.ft.313.2cu.in. 
523.6  cu.  yd. 

Art.  934. 

2.  10  ft. ;  15  ft. ;  and 

20  ft. 

3.  24  ft. ;  32  ft. ;  and 

40  ft. 

Art.  936. 

1.  13.228  ft.  edge- 
2315.03  cu.  ft.  vol. 

2.  11  ft.  7  in. 

3.  1494.257  gal. 

4.  $5.46. 

5.  576  ft. 

6.  14.42  in. 

7.  40  sq.  ft.  7f '. 

8.  1  cu.ft.  vol.of  cube 
1  cu.  ft.  659  cu.  in. 

vol.  of  sphere. 
P.  9  lb. 

10.  5  hr.  26.4  min. 

11.  12  ft.  6.79  in. 

12.  53.855  bu. 

Art.  937. 

2.  99.144  gal. 

3.  120.09  gal. 


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